Dance: Befitting the Korean origins of her inspiration, her dance is Run Devil Run by Korean pop group Girls' Generation. A side by side here.
Akali
Skins: Stinger Akali resembles The Bride from Kill Bill, Crimson Akali looks like Elektra from Daredevil, and Silverfang Akali looks like Smoke from Mortal Kombat 9. On top of it all, her classic skin strongly resembles Jade from Mortal Kombat.
Skins: Sherwood Ashe gets her name and primary colors from Robin Hood. Woad Ashe looks like Guinevere in the 2004 film of King Arthur.
Blitzcrank
Skins: iBlitzcrank's design is much like the Apple Macintosh, and was released alongside LoL's beta client for Macs. The name itself is a reference to Apple's line of electronics which have a similar naming scheme (e.g. iPhone, iTouch).
Ironically enough, when iBlitzcrank dies he suffers a Red Ring of Death; a reference to the Xbox made by Microsoft, Apple's main competition.
Brand
Skins: Apocalyptic Brand looks like he just stumbled out of the Fallout world.
Brand's origin seems to have been born out of Surtr, with Brand's body being an essential viking, his appearance being similar a common description of Surtr (blackened skin with lava crackling in it), and his claims that him burning the world will renew it, as would follow Surtr's role in Ragnarok. Regretably, Brand has no Flaming Sword.
Cho'Gath
Skins: The splash art for Jurassic Cho'Gath, with Cho'Gath in a museum-style building with a large banner hanging overhead, is a reference to the Jurassic Park film.
Some people can't help but look at his design and think "Violator." Other people think he looks a lot like a Carnifex. In early beta, his title was "The Red Terror", a type of Tyranid.
Corki
Quotes: "This is Major Tom to Ground Control!" His /joke is "Lima Oscar Lima", which is NATO phonetic for LOL, a shout out to the internet slang for 'laughing out loud'. Or, y'know, League of Legends.
Possibly a coincidence, but the fact that Sheen is a common powerful item on Corki makes this rather uncanny.
His energy bow looks a lot like the bow used by the Quincy in Bleach.
Riot Games's advertisement on YouTube for his Pulsefire skin is a love song to Konami's Zone of the Enders: The 2nd Runner. Watch the PS2 video game intro trailer and have fun finding inspirations/similarities.
Fiora
Dance: Fiora does her best impression of Fred Astaire dancing to Puttin' on the Ritz.
Skins: His Special Forces skin brings Captain Price to mind.
Garen
Skins: His 'Sanguine' skin was explicitly made to evoke a Blood Angel Space Marine from Warhammer 40000, his Commando skin evokes the Imperial Guard, and his Dreadknight skin is a few letters and a space away from being a Death Knight.
Steel Legion Garen is explicitly a Space Marine. Sadly, there is no chainsword.
Quotes: The first sentence in one of his jokes ("Summon me and you'll like the way you look. I Garen-tee it. Get it?") is a reference to Men's Wearhouse commercials.
Quotes: One of his lines "Today is a good day to die!" could be one to Star Trek, but possibly instead to Sitting Bull or Crazy Horse. He says a line from The Charge of the Light Brigade verbatim: "Ours is to do and to die." He also says "For my father, the king.", a line of Prince Arthas from Warcraft 3. As one of the developers worked on Defense of the Ancients: All-Stars, a Warcraft 3 custom map, this was quite certainly intentional.
Skins: His hockey-themed skin is called The Mighty Jax. PAX Jax, which was released for the Penny Arcade Expo, appropriately wields a cardboard tube and has a Pac-Man logo on his clothes.
His name and character are a reference to Garet Jax, the armsmaster who was an expert with all weapons, in Terry Brooks' The Wishsong of Shannara.
Jayce
Skins: With his Full Metal skin, he could either pass as Iron Man or Samus Aran at a glance.
Skins: Phantom Karthus wears a mask like The Phantom of the Opera, and Pentakill Karthus strongly resembles Gene Simmons from KISS, as well as King Diamond to a lesser degree.
A somewhat elaborate example; Karthus' innate passive's icon, Death Defied was originally a picture of a sheep crossed out. One of the most powerful items in DotA is Guinsoo's Scythe of Vyze, which turns people into sheep. Guinsoo uses Karthus' icon in the official forums. You make the connection.
Kassadin
Skins: His Harbinger skin is somewhat reminiscent of Emplate.
Katarina
Dance: Taken from the famous dance scene at the end of Napoleon Dynamite.
Skins: Deadly Kennen is a fairly obvious Deadpool costume. Swamp Kennen is Yoda.
Kha'Zix
Skins: He's heavily based off of a Xenomorph, while his ability to assume the most desirable characteristics of the prey he devours is right out of the Kroot and Tyranid playbooks.
Kog'Maw
Dance: Does the kicking dance done by the from from the animated classic One Froggy Evening.
LeBlanc
Skins: Her Wicked skin resembles Cruella de Vil of 101 Dalmatians.
Lux is largely based on Marisa Kirisame from Touhou. Her ultimate is named after Marisa's Final Spark (a variation on her trademark spell known as Master Spark), which is also a giant rainbow laser beam. Lux also calls herself an "ordinary witch," while Marisa is The Ordinary Magician, and both are known for copying spells from other people. Finally, Lux's Sorceress and Spell Thief skin may be based off of Marisa, most notably the witch's hat and broom on sorceress and the similarity of the outfit and wand of Spell Thief to Marisa's outfit in UFO.
Skins: Her Commando skin is based on the Imperial Guard from Warhammer 40000. Bonus points for having a wand that's pretty much a lasgun on each end.
Skins: His Chosen skin is pretty much a Jedi knight, complete with blue, green, or red light saber depending on how much his attack damage is increased by his Wuju Style ability. His Headhunter skin is based on Predator.
Dance: A brief segment of the dance at the end of Napoleon Dynamite.
Maokai
His saplings look, move, and sound quite a lot like Pikmin.
Miss Fortune
Quotes: Several of her quotes are references to Han Solo.
Mordekaiser
Abilities: Each one is inspired by a heavy metal song.
Quotes: He's obviously a walking reference to 300, and both his taunt and his joke reference the movie.
Poppy
Skins: The face paint on Noxian Poppy looks an awful lot like the members of KISS. Lollipoppy looks just like one of the Oompa Loompas from Willy Wonka & the Chocolate Factory.
Skins: King Rammus changes his color scheme to closely resemble Bowser. Ninja Rammus is a reference to the Teenage Mutant Ninja Turtles, specifically Raphael.
Rengar
Skins: He's basically a walking homage to Predator. His Headhunter skin makes this more apparent.
Dance: His dance is the Haka, of rugby All Blacks fame. This may further be a reference to the similar looking Charr race in Guild Wars, as they also use the Haka as their dance animation in Guild Wars 2. Very possible as the Lead Champion Designer Morello worked on Guild Wars 2, and Rengar's release very closely coincided with Guild Wars 2's.
Riven
Skins: Her Redeemed skin is based off of Meliadoul.
Skins: Darkflame Shyvana's dragon form bares a resemblance to Alduin.
Dance: Does the same dance as Japanese idol group AKB 48 in the music video for their song RIVER.
Singed
Skins: Augmented Singed is a very obvious reference to Bane, who was the main villain in The Dark Knight Rises, which was released at about the same time as the skin was.
Skins: Her Warrior Princess skin is Xena, her Spectacular skin is Wonder Woman, and her PAX skin is based on TRON, her huntress skin resembles Zhu Rong as well as giving her a boomerang (her preferred weapon until 6).
Sona
Skins: Arcade Sona's ultimate (which sends out a wave causing enemies to dance) resembles a Dance Dance Revolution screen, with a flow of arrows pointing in different directions.
It's generally accepted that her standard skin was inspired by Hatsune Miku.
Swain
Skins: His form while using his ultimate is awfully reminiscent of a Lord of Change. His Tyrant skin makes this even more obvious.
Abilities: One of them is called Nevermove, a pun on Nevermore that was only supposed to be used during development but ended up sticking.
Skins: Her Vindicator, Aristocratic, and Dragonslayer skins skins are likely references to Devil May Cry, Bayonetta, and the classic Dragoon-class from Final Fantasy, respectively.
Dance: Appears to look a little like the poses the Ginyu force pulls off, it might be a Gunkata though.
Her character and background draws heavy inspiration from Batman. She even has bats flying around her when activating her ultimate!
Abilities: One of his skills, Tides of Blood, may be a reference to an old MOBA-styleWar Craft 3 custom map of the same name. Said map had a "Blood Mage" hero who had more-or-less the same gimmick as Vladimir.
Bares a strong visual resemblance to Majin Buu, and his backstory is equal parts Superman and Hellboy. His classic look is also referred to as Flubber.
Zed
Bares a lot of resemblance to Power Rangers Lord Zedd, with a little bit of TMNT's Shredder thrown in.
Skins: Shurima Desert Zilean looks a bit like Gwonam.
Other
NPC's
If two or more champions perform their dance emotes in front of Vilemaw, the giant spider boss of Twisted Treeline, he'll lift his legs up as if surprised, look around to make sure no one's watching, and start doing the Carlton Dance.
The two new shopkeepers on the Howling Abyss map will occasionally reference movie quotes:
The icon for Mastermind in the support mastery tree is a reference to the box art from Deus Ex.
Items
Youmuu's Ghostblade is one letter off from Youmu from Touhou, who also wields a ghostly sword. It is a katana with sakura petals flying around it.
Wriggle's Lantern shares a name with a firefly in Touhou. The official statement is that it was named after an adjucator called Wriggle, but behind the scenes we know that (former) employee Shurelia is a big Touhou fan.
Trinity Force is the most expensive item in the game, created by combining three different mid-level items into one hybridized one. Both its name and its in-game icon are a shout-out to the Triforce from the Zelda series.
The Brutalizer item costs exactly 1337 gold in total.
Rylai's Crystal Scepter is named after the lore name of Crystal Maiden from Defense of the Ancients: All-Stars. The item icon is almost identical to the one for Rylai's Brilliance Aura ability.
The Manamune is highly reminiscent of the Soul Calibur sword from the first few games.
Its name is also derived from Masamune, a legendary Japanese swordsmith. Its upgraded form, Muramana, is the same case; Muramasa was also a famous Japanese swordsmith.
Atma's Impaler references the "Atma Weapon" mistranslation of Final Fantasy VI.
Guinsoo's Rageblade is half shout out and half internal homage, as Steve "Guinsoo" Feak, one of the big names at Riot Games, used to develop Dot A Allstars before passing it on to current developer Icefrog. For that matter, a number of champions and items take their moniker from Riot developers' screen names. But the most convoluted Easter Egg is the fact that Karthus's passive used to have a sheep as an icon. This referred to "Guinsoo's Scythe of Vyse," an item from Dota which conferred Baleful Polymorphing, and was snuck in because Karthus is Feak's favorite Champion.
Meta
An update for a more exact queue status now shows your place more precisely. Initially, when the queue was long enough, it just reports your current place in the waiting line as Over 9000. The queue is now more exact (Over 20,000), but lost the reference.
The in-game chat displays kill streaks using Unreal Tournament lines, such as "Killing Spree!" "Dominating!" and "Godlike!"
One example from the Journal of Justice that coincides with Getting Crap Past the Radar is Janna being caught in the middle of a scandal that's heavily implied to involve a sex tape, and trying to pass it off as boosting her magical reserves, bringing to mind the notorious "Mana transfer" in Fate/stay night.