You have probably heard some stories influenced by these myths, although you might not realize it. These are the tales that The Fair Folk come from.
There are two main (surviving) strands of Celtic mythology: Goidelic (Irish/Scots/Manx) and Brythonic (Welsh/Cornish/Breton). While they share many tropes and have certain figures in common, they do not really overlap; each has its own unique stories. They are further split into 'Cycles' (Ireland) and 'Branches' (Wales). Mainland Europe's Celtic traditions were mostly lost due to invasion and assimilation of Celtic populations in their conquerors' own societies (mainly The Roman Empire and Germanic tribes). The cultural taboo against consigning knowledge to writing certainly didn't help.
In a nutshell: before people came to the archipelago we now call the British Isles, a race of intelligent magical non-humans calling themselves (in Irish, anyway) the Tuatha Dé Danann ("the children of the goddess Danu") lived there. With the arrival of people and their permanent settlements, the Tuatha Dé Danann continued to muck about in the lives of people, but retreated to the Otherworld, their home world, a world still reachable through places such as fairy forts or fairy burrows. (Interestingly, the "gateways" identified in Celtic stories would not infrequently turn out to be archaeologically significant sites dating to the Neolithic period.)
Celtic mythology also includes Scottish, Breton, and Cornish stories.
See Irish Names for pronunciation help.
Works that are part of Celtic mythology that have their own pages:
Awesome, but Impractical: The Gae Bolg, a spear owned by Cú Chulainn, which pretty much guaranteed victory against any enemy. However, it had to be blessed in a stream before use and thrown from the foot. On impact it would extend barbs down every blood vessel of the target's body, killing them... and making it impossible to use again without cleaning the corpse off first. This made it entirely useless in large-scale combat, though Cú Chulainn sometimes used it in duels.
Also the Spear of Lugh, one of the Four Treasures. While incredibly powerful, it had to be kept in a bath of blood to avoid it bursting into flames and draining the blood of everything around it. Imagine how that would have affected the household where it was kept. Fortunately poppy juice could keep it sedated, and another of the Four Treasures was a cauldron which could "satisfy any hunger".
Awesome Mc Cool Name: Cu Chulainn means "Hound of Culann" (explained under Martial Arts and Crafts below).
And nowadays, there's Fionn MacCumhail. How do you pronounce that? Finn McCool.
Bad Ass: Several, with Cú Chulainn being the best example.
Badass Army: The Fianna, famously lead by Fionn MacCumhail.
Badass Transplant: Nuada Airgetlam. Airgetlam means silver hand. It was a literal title.
Because Destiny Says So: When Cú Chullain was still a child of six he overheard a druid prophesying that anyone who took up arms on that day would become the mightiest hero of Ireland. So he immediately rushed off to the king and asked for a sword... But also, see The Chosen One below.
The Berserker: Cú Chullain. After one battle they tried to calm him down by mobbing him with topless women. When he turned his face away in embarassment, they grabbed him and threw him into a barrel of water. The barrel exploded. They tried it again and the water boiled away. Third time was the charm, since the water only became somewhat warm.
Cú Chullain, when he's not Crazy Awesome or outright terrifying. He was so insanely attractive that the men of Ulster wanted him married off as soon as possible to keep him from stealing their wives and seducing their daughters.
There's also Naoise (who is described to Deirdre, his eventual lover, with the "hair as black as ebony, skin as white as snow, lips as red as blood" description familiar from "Snow White"), Midir, Taliesin, Gwydion... The Celts sure like their pretty guys.
Body Horror: Cú Chullain's "warp-spasm" causes his legs to turn backwards, his temples to swell, one of his eyes to fall out of its socket while the other is sucked down into his head, his mouth to split open, and his hair to twist into spikes. There's a reason people were scared of this guy.
The Chosen One: Cú Chullain was a preternaturally strong berserker from early childhood, surrounded by prophesies and geasa. His actions in taking arms (see Because Destiny Says So above) was his statement that he was perfectly OK with being the Chosen One.
Continuity Snarl: Lots of little snarls crop up when you get past the general idea, what with the lack of written records for about a thousand years, the constant invasion by the Romans/English/Norse, and the replacing of pagan beliefs with Christianity.
Cool and Unusual Punishment: One version of the curse upon Ulster's men came about when a pregnant woman on the road was going into labor, but none of the men passing by would offer any assistance, so she cursed the whole land so that the men would experience the pains of labor every year. In the Táin Bó Cúailnge, Connaught times its raid for when the warriors of Ulster were crippled by that time of the year.
In the more canonical version, the punishment is imposed by Macha after she's been forced to race a horse while pregnant. She wins the race, gives birth to twins, and then dies, giving the curse as she does that the men of Ulster will be as weak as a woman in labor whenever they are in the deepest need.
When Rhiannon is accused of eating her newborn baby (the maids framed her), Pwyll doesn't want to kill her. So instead, she has to carry visitors from the courtyard to the castle hall on her back. For seven years.
Cool Horse: Grey of Macha and Black of Saingliu, the twin steeds of Cúchulainn, and Morvarc'h, the coal-black steed of the Breton king Gradlon.
Pryderi and Rhiannon's horses.
Embarr, the white horse that belong to Niamh (and later lent to Oisín) which could run over the land and water.
Cranial Processing Unit: The Celts saw the head as the location of the soul. Which is closer to the truth than a lot of religions managed - the Egyptians saw the brain as mush to be disposed of while they preserved your more important organs.
Dark Action Girl: Aoife, Medb, Macha, and especially the Morrigan. They're wrathful, petty bitches, but like hell if they aren't Badass.
It's easy to find bowdlerised versions of the stories aimed at younger readers. Probably the most commonly heard child-friendly story is Fionn's building of the Giant's Causeway, and subsequent outsmarting the Giant that came to fight him. Not a drop of blood spilled, and Fionn dresses up as a baby as part of the trick.
Deadpan Snarker: Diarmuid occasionally shows himself to be one in his interactions with Grainne. At one point, Grainne stabs Diarmuid in the thigh after they have a tiff. When Grainne later asks for a knife to cut some meat, Diarmuid tells her to search the sheath she last put it in, casually pointing to his still impaled thigh.
Death Amnesia: Cerridwen's cauldron brings back the dead but, because it's forbidden for the living to know anything about the afterlife, the reanimated dead cannot speak.
Did You Just Punch Out Cthulhu?: The Tuatha Dé Danann didn't leave this world and make way for humans by choice. Mankind went to war against them, won, and drove them out so that they could claim their land. It is the only time in any mythology where humanity kicked the gods' collective butts.
Also, the Tuatha De Dannan are exclusive to the Irish Portion of the Celtic mythology. The Welsh, for example, have the Children of Lyr and Dôn as their supernatural clans.
Divine Conflict: The conflict between the Tuatha De Danann and the Fomorians over the ownership and right to settle the island of Ireland from Irish Legends.
Double Speak: In Wales, the fairies were called "The Fair Folk" (Tylwyth Teg) because you could praise them all day long and they would pay you no mind. Similarly, you could safely refer to the dangerous, unpredictable magical people as "Lords and Ladies", "the good neighbours", and a host of other polite euphemisms—they had prickly tempers, and an insult was an invitation for anything ranging from dead livestock to a swift death.
Droit du Seigneur: Conchobar was actually obligated to sleep with Emer before Cu Chulainn. He was understandably too scared to actually sleep with the wife of a man known to go into crazy, murderous rages, so he simply shared a bed with her while Fergus and Cathbad stayed in the room to confirm that nothing actually happened.
For the Evulz: Efnysien mutilates all of the Irish horses, seemingly for this, and then crushes the skulls (described in detail) of the soldier's hiding in bags with little to no provocation.
Geas: Trope namer. Unlike in modern times, it wasn't really seen as entirely a bad thing. If you were under a geas and fulfilled its terms, it made you stronger spiritually. It was still the downfall of numerous heroes, though.
God Save Us from the Queen!: The Morrígan ("Phantom queen" or "Big Queen" or "Queen of Death" or "The Queen"), Irish goddess of war, death, and other fun pastimes. Overall, her morality falls under Blue and Orange Morality rather than outright evil, but for humans they don't notice the difference.
Medb/Maeve of Connaught, who raised an army and invaded Ulster just so that she could steal a magic bull to match the one her husband owned.
It's about the primacy of female or male lines of inheritance, plus cattle raids were a show of power back then. Although some say it's about Medb and Ailill being irresponsible jerks.
The Hecate Sisters: The Irish goddess of war known as the Morrigan was often (but not always) depicted as a triple entity; the most common combination is the Badb, Macha and Nemain, but other accounts name Fea, Anann, and even Erinn (lit. Ireland). They can be shown as Mother, Maiden and Crone, or all one facet.
Hijacked by Jesus: The myths were not written down until after most Celts had converted to Christianity — as a result, gods were converted to kings and heroes, and millennia-long curses are broken by priests. It's all pretty muddy.
For instance, it's unknown exactly how much of what we know of St. Brigid was an actual historical figure, and how much was stories originally from the Celtic goddess Brigid.
The Fair Folk went from... the Fair Folk, who were all right if you didn't offend them, to evil spirits who couldn't stand church bells/crosses/a priest.
Infinity+1 Sword/Stock Weapon Origins: The Spear of Lugh, which spouted gouts of flame and was so thirsty for blood that it wielded itself with little help from the owner. Lugh had to keep it submerged (in either milk of the poppy or blood, depending on the telling) to keep it under control.
Ditto the Claíomh Solais or "Sword of Light" (not that one), the sword of Nuada before he was slain. Excalibur on crack is a decent way to describe it. It made a reappearance in The King of Ireland's Son.
Tritto Caladbolg (possibly an earlier version of Excalibur), the sword of Fergus Mac Róich, which even Cú Chullain was afraid of. Literally "Hard/Solid Lightning," it made a pretty rainbow trail when swung... and could kill gods. No, seriously, it COULD kill GODS. He missed hitting an enemy once and the ensuing strike lopped the top off of three adjacent mountains!!!
Finally, Excalibur, i.e. the bar-none most famous sword in Welsh-Celtic MythologyIN THE WORLD. Not a whole lot is consistent about it, but two things are always present: one, when drawn, it'll burn out the eyes of anyone in line of sight of the blade besides the wielder; two, it could basically cut through god-on-anything like a lightsaber through half-melted butter that wasn't magical as well - and even then, it'd better be damn strong, like the Grail Sword. Finally, its scabbard had defensive powers varying between "the bearer's wounds don't bleed" to "the bearer becomes completely invulnerable".
Kick the Dog: To get back at Cuchulainn for knocking her up and then marrying Emer instead, Aoife sent her son out into the world with two conditions: Challenge every warrior he meets, and never ever ever reveal his name. Naturally, when Cuchulainn gets challenged by some kid, cue the Curb-Stomp Battle. Then Cuchulainn notices a really familiar-looking ring...
Losing Your Head: Bran the Blessed's head stayed alive after his allies struck it off his dying body at his instruction. It was buried in the White Hill of London, supposedly the spot where the Tower of London now stands, and the story goes that Britain can never be conquered as long as the head is there. Though a tradition says that Bran's head was dug up by King Arthur, who wanted to be solely responsible for protecting Britain from invasions, and that would explain why there are so many Anglo-Saxons in Britain, let alone the Norman invasion.
Martial Arts and Crafts: Cu Chullain (childhood name "Setanta") as a child killed a massive guard dog belonging to the smith Culann by smashing a sliotar (a ball used in hurling, which is like a cross between hockey and rugby)) down its throat with his hurley. He repaid the smith by acting as his guard dog until the original dog's puppies were fully grown. This is where he gets his name—Cu Chullain means "Hound of Culann".
Mysterious Mist: The féth fíada or fairy mist (also ceo druidechta, druid mist), which was used by the Tuatha Dé Danann to conceal themselves from observers. According to the Lebor Gabála Érenn, the Tuatha first arrived in Ireland wrapped in this mist, after travelling over the sea in it. In stories set in later times, druids and even Christian saints are reported to make themselves invisible with magic mist.
No Man of Woman Born: Cú Chullain was the sole defender of Ulster owing to a curse which stated that all fighting men of Ulster would be crippled for nine days and nine nights when they were needed most. While Cú Chullain was the champion of Ulster, he was not an Ulsterman (and, technically, still a boy), and was thus exempt from the conditions of the curse when Medb invaded.
Numerological Motif: The number three tends to show up a lot, from the above mentioned triple goddess to the famous symbols you see on various Celtic paraphernalia. Seven and nine are also common.
One Steve Limit: Medb obliterates this trope, spectacularly. A druid tells her that one of her sons will one day kill Conchobar (one of Medb's many ex-husbands and king of Ulster). When Medb asks which one will do the deed, the druid simply responds, "Maine". Though she had seven sons, none of them were named Maine, so she hedged her bets and renamed all of them Maine. One of them eventually does kill Conchobar, though it turns out to be a different Conchobar than the one she was hoping for.
Only Known by Their Nickname: Cú Chullain's birth name was "Setanta." Presumably most people know it, but there's a lot more stuff named after Cú Chullain.
The other famous Irish hero, Fionn Mac Cumhaill, was given his first name after his hair turned white (Fionn means "Blond" or "fair"). His birth name was Deimne.
Overly Long Name: Fer Benn Bruach Brogaill Broumide Cerbad Caic Rolaig Builc Labair Cerrce Di Brig Oldathair Boith Athgen mBethai Brightere Tri Carboid Roth Rimaire Riog Scotbe Obthe Olaithbe, better known as the Dagda.
Paper-Thin Disguise: Cú Chullain, pursuing Connaught's warriors to retrieve the white bull of Ulster, challenges one to a duel. When he refuses to fight a "beardless boy," Cú Chullain (being only seventeen at the time) runs off, finds a goat, shaves some of its hair, and weaves a fake beard to wear before making the challenge again.
Sadly Mythtaken: The supernatural beings of the Otherworld (Sidhe, deities, or spirits), are often called fairies—without the modern connotation of "adorable tiny winged people". That innovation came from the Victorians. Some modern fantasy authors use alternate spellings such as "faery/faerie" to convey that they mean the Darker and Edgier version of The Fair Folk.
Serious Business: Storytelling. The humble bard had incredible power in the Celtic tradition, because of his ability to tell stories. The worst thing that can happen to a mighty warrior is not to be defeated in battle, but to have a bard compose a satirical poem about him. A superior opponent can merely kill you. A well-written story about how much of a loser you are will be told and re-told until the end of time.
Slap-Slap-Kiss: Diarmuid and Grainne have this initially after eloping: they fled from Fionn and the Fianna and were on the run for weeks, but despite them being madly in love, Diarmuid refuses to have sex with her, out of loyalty to Fionn, who was to be married to Grainne before she fell in love with Diarmuid. One day, while on the run Grainne steps into a puddle and water splashes between her thighs. She mocks Diarmuid, saying how even water in a puddle is braver than he is. His noble pride hurt, Diarmuid finally makes love to Grainne.
Sunken Lands: There are a number of kingdoms and cities drowned underneath the waves: the Welsh kingdom of Cantre'r Gwaelod, the Breton city of Ys (or Ker-Ys), the Cornish kingdom of Lyonesse, and—in some versions of the Arthurian legend—Avalon.
Talking in Bed: The beginning of the Irish story Táin Bó Cúailnge, "The Cattle Raid of Cooley."
Unusual Euphemism: Lots of people refer to "The/a garden of the Morrigan" or "the Morrigan's harvest." While she IS a fertility goddess, her "garden" and "harvest" are actually battlefields and the souls / heads / eyes of the dead. This possibly refers to her sacred birds, the corvids, who eat carrion.
It might also refer to the fact that battlefields become quite verdant after a few years.
Also, "Macha's Acorn Crop" is a poetic way to describe the piled-up heads after a battle.
When she's trying to get a valuable bull, Medb promises his owner the "friendship of her upper thigh."
If The Wooing of Emer is to go by, the Monty Python crew was far from the first to refer to breasts as tracts of land.
Cu Chulainn: When I said, "Fair is this plain, the plain of the noble yoke," it was not the plain of Bray that I praised then, but the shape of the maiden. For I beheld the yoke of her two breasts through the opening of her smock, and it is of that I said, "plain of the noble yoke," of the breasts of the maiden.
In Tir na nÓg you experience time normally, but don't age it until you leave. Which isn't a pretty sight.
Works substantially derived from (or outright retellings of) Celtic mythology:
Anime & Manga
Code Geass contains some references to Celtic Myths, such as the Geass (derived from the geis/geas). It also heavily references Arthurian Legend, as well as Lelouch's death having shades of King in the Mountain, due to Wild Mass Guessing over his death/possible survival. It helps that The Empire refers to itself as Britannia and makes heavy use of the legends and actual history of the British Isles.
Durarara!! also derives significant plot points from Celtic mythology, such as Dullahan Celty.
The eponymous character of Pat Mill's Sláine comics is almost a straight expy of Cú Chullain. Same battles with Queen Medb, same barbed death spear, same horrible body-warping berserker rage. The whole series mixes Celtic Mythology with some Conan the Barbarian, some Lovecraftian horror and a titch of Neo-pagan spirituality. Season with a punk rock aesthetic and serve.
In a Marvel ComicsThor miniseries Thor: Blood Oath, Thor and the Warriors Three are tasked with retrieving a spear called Slaughter from the Irish gods. The spear appears to be a combination of the Spear of Lugh and Gae Bolg. It's kept in a cauldron of blood to prevent it from killing anyone who happens to be nearby and is wielded by Chulain, though it doesn't have any extending barbs.
Franco-Belgian Comics have had a large influx of Celtic Myth and Folklore-inspired titles in the Turn of the Millennium that is still going strong; in fact, one of the largest "bandes dessinées" publishers, Soleil, has a whole sub-imprint just for those. This is hardly surprising given that before Roman conquest France and Belgium were Gaul, the largest and most populous of the celtic lands, so it can be seen as going back to their roots (also a ready-made source of inspiration to help stay competitive in the face of manga, which is extremely successful there).
Film - Animated
The Secret of Kells, being a film made and set in Ireland, is naturally filled with Celtic mythological elements as well as Christian ones. Aisling is stated by Word of God to be one of the Tuatha Dé Danann.
Traci Harding's Ancient Future Trilogy, and its sequel trilogies.
The Dresden Files being a Fantasy Kitchen Sink, involves the Fae (make no deals and eat none of their food, just like the legends, as well as changelings, being the child of a fae and a mortal), Excalibur, and Merlin has been brought up on several occasions.
C. S. Lewis' The Voyage of the Dawn Treader takes much inspiration from the ancient Irish genre of immrama, "sea-voyages" — stories about fantastical seafaring expeditions into the unknown Western ocean, beyond which lies the paradisical Tír na nÓg, or Land of the Young; only in Voyage of the Dawn Treader, it's the Eastern sea, at the end of which lies Aslan's Country.
Artemis Fowl, written by the Irish Eoin Colfer features fairies, who, like in the myths, retreated underground and built a highly technologically advanced civilization that fears humanity and - wait...
The Goddess Danu's protection of the Earth and fairies is a major plot element in the last book.
Patricia Kennealy-Morrison's Keltiad series is Celtic MythologyIN SPACE!! — St. Brendan led his people on a great migration to another planet.
Fiona Patton's Tales of the Branion Realm series has elements of Celtic Christianity, particularly the Kingdom's founders, who are named Bran Bendigeid and his sister Braniana after characters in Welsh mythology.
The Sevenwaters Trilogy is set in history but draws much of inspiration from Celtic Mythology. Interestingly though, the plot line is based on a German fairy tale.
At Swim-Two-Birds by Flann O'Brien has Finn Mac Cool and Mad King Sweeney appear as characters.
The Wanderings of Oisin, a narrative poem by William Butler Yeats, is an retelling of Oisin's adventures in Tir nan oge.
Salvatore Doni and Duke Voban have both received Authorities from figures mentioned on this page in Campione!. Nuada and Balor respectively.
The 1998 miniseries Merlin, which includes many elements of Celtic Mythology rarely seen in retellings of Arthurian Legend.
.hack features Macha and Morganna (The Morrigan) as digitalgoddesses. Lia Fail, the Tuatha Dé Danann's stone of destiny is one of the Root Towns of The World. Three characters are known collectively as the "Descendants of Fianna". And Crennuos is also used by CC Corp for the The World's backstory in .hack//G.U.
A substantial portion of the Shadow Paladin clan of Cardfight!! Vanguard are named after Celtic mythological characters, such as: Cursed Spear Revenger, Diarmuid; Darkness Maiden, Macha; Skull Witch, Nemain; Dark Mage, Badabh Caar; and Witch of Cursed Talisman, Etain. Interestingly, Macha, Nemain, and Badabh Caar are clasically the three sisters of the Morrigan, and in-game all have effects based around incresing your card advantage, probably so you can simply use it up to pay the costs of powerful effects.