As of Generation VI, there are over 700 Pokemon in the world, along with thousands of trainers, the anime incarnations of specific mons and characters, and various AU games of Pokemon. When you have this many Loads and Loads of Characters, naturally, some will stand out better than others.
Spinoff game Pokémon Colosseum has Miror B. Crazy Awesome, literal Disco Dan with a Pokeball-shaped afro as the first villain team boss. He never cares about his duties as a boss of a crime syndicate and prefers to practice his dance moves. He also comes with his latino sounding battle theme and got a second appearance in the sequel where you could battle him as often as you want (or until you got all the Shadow Pokémon, whichever came first).
The homeless fat guys that usually hang around starter towns and rave about technology. No seriously, just listen to starter town tunes on Youtube, and count how many times you see someone comment on the fat guy standing there. He has become epic, in his own right.
In a similar vein, the guys that greet the players in every Gym with "Yo! Champ in the making!" Until the 4th gen... It's unclear if it's a different guy each time or the same one, but either way, he's as beloved as the fat technology guy.
Leaf, the Distaff Counterpart to Red, hasn't appeared in any game besides her debut and is mostly forgotten by Game Freak nowadays but has quite the following. People are always begging for her to cameo or be fightable in games, potentially being a higher level than Red too.
Hilbert and Hilda of Pokémon Black and White are probably the most popular protagonists ever since Red and Leaf due to their overall Bad Ass designs (Hilda being a Tomboy) and the fact that they're teenagers rather than children like the previous playable heroes. Not to mention that Hilda is very frequently shipped with N (see above).
Misty and Brock are this in the games due to their anime and Pokémon Special counterparts. In the games themselves they're just early gym leaders, with Misty having a slightly larger role than most gym leaders in the Johto games. However, thanks to their counterparts, they've become amongst the most popular characters in the franchise.
Misty has not been a main character on the anime since 2003, yet she's still popular among the fanbase, even among viewers who started watching years after she left the show.
Similarly, while fans may argue about which generation is the best, there's no denying that Professor Oak is the most iconic and popular Pokémon Professor in the franchise.
In Japan, the Subway Masters Ingo and Emmet from Black and White are ridiculously popular, ranking second and third on a popularity poll despite not having any major role in the story.
They've gotten so popular that they've gotten quite a lot of official merchandise, more than even the protagonists. It's surprising considering they have no real personalities and are barely in the games.
Cynthia, the first female champion from Diamond and Pearl, has also become quite popular. She's managed to appear in HeartGold and SoulSilver and Black and White, which isn't bad considering that Pokémon has non-linear sequels.
Professor Sycamore from Pokemon X and Y has become the most popular professor since Oak due to being a incredibly handsome womanizer and the first professor to actually battle you.
Pokémon - Generation I
Mewtwo is extremely popular; easily the only other Pokémon from Gen 1 whose popularity can compete with Charizard's. It has a massive following, particularly among the older male demographic. Points also go to its dark and mysterious backstory. Being the original ultimate Pokémon definitely helps (although you'll find a dedicated legion on the internet ignoring Arceus all together and seeing Mewtwo as still the ultimate mon), as does being the star of the very first Pokémon movie.
Cubone and its evolution Marowak are remembered partially because of their interesting design, use of bones as clubs, tearjerking backstory, and the fact that they're the only Pokémon in G1 to have actual story relevance (the next closest to having to do with the plot was Snorlax, who simply was a Broken Bridge).
Pidgey. Still liked by many fans despite its shortcomings for being the original Com Mon bird. Perhaps it's the fact that it may have been the very first Pokémon to actually be caught by the first Pokémon trainers, or simply because it's cute.
Similarly, Ratatta is liked due to the top percentage meme mentioned above, and the fact that it's also the center of the "FEAR" strategy.
Meowth, probably because of its appearance in the anime. Almost the same case as Pikachu, but toned down quite a bit.
Vulpix/Ninetales. You don't have to go far to see the MANY fan-fics that claim this to be Naruto's favorite Pokémon. Add in the fact that the fox Pokémon were given Groudon's Drought ability, bringing about its usage in Sunlight teams.
Gengar has always been considered one of the more powerful and cool looking ghost-types. In Gen I, it was the only fully evolved Ghost type. Even with the introduction of other ghosts, it has always managed to remain in the higher tiers. Its Mega-evolution is no joke either due to its ability becoming Shadow Tag.
Magnemite gained in popularity over the years. First, becoming a steel-type in the second generation, which greatly increased its usability. Then, it gained an epic secondary evolution in Generation IV which made it a powerful Steel-type special user. In Pokémon Black 2 and White 2, Magnemite were used in the Magnemite Coil, a form of Emergent Gameplay, in Join Avenue. note An area in the game lets you set up shops where you get services not offered elsewhere, like leveling up your Pokémon, buying berries, or buying common items cheap in bulk to sell at higher prices in other stores. If you trade Pokemon with other people, NPC's modeled on these trainers will arrive at the area and interacting with them will either let you set up the shops you want or "level up" the already existing shops, thus expanding their services. Cue Magnemite: being one of the earliest Pokemon available in the game, if you catch one and go to the Global Trading Station at any Pokémon Center, you can trade Magnemites of any kind with trainers from all around the world. This way, you can get more NPC's to level up your stores and use their services much sooner than you would otherwise.
Pinsir was another well-liked bug Pokémon from the first generation, but lost a lot of support over the years due to being overshadowed by the likes of Scyther (Scizor), Heracross, and Scolipede. Generation VI, however, finally throws Pinsir a bone and bestows upon it a Mega-evolution, putting Pinsir back on the map for the fan-base.
Eevee simply for being the reason Eeveelution exists.
Jolteon for being one of the few Pokemon that could take down Psychic types in the first generation due to its blazing speed. Competitively, it has the honor for always being in the Overused tier prior to Gen VI.
Flareon gained quite a following other the years mostly for being the underdog of the Eeveelutions; getting constantly screwed over for never receiving any moves to take advantage of its monstrous attack stat until Generation VI. And even then...it still suffers a lot of problems.
Vaporeon as an Eeveelution and as a highly competent Bulky Water competitive use Pokémon.
Dragonite, Generation I's pseudo-legendary and the only Dragon type at the time. Dragonite waned in popularity as the generations went on due to being overshadowed by Salamence and Garchomp, but once Generation V bestowed upon it the hidden ability, Multiscale, its usability skyrocketed.
MissingNo, in a mixture of Good Bad Bugs and Memetic Mutation. The fact that his presence glitches the world in a hilarious manner and replicates the item in the 6th slot (useful for Master Balls and the like) helps a lot, too.
Mew, the kitten-like predecessor to Mewtwo, is popular due to its rarity (being an event-only Pokémon, unless glitches are used), cuteness, and overall versatility, as it can learn every single move that can be taught with an item in the game.
The baby Kangaskhan holds has a large popularity. If someone is creating their own fan generation and Pokémon, the baby is a popular choice for a Pokémon.
Scizor. An evolution to a Pokémon that was already a darkhorse. Gains a steel typing and epic looking design to make it even more Badass, and then gains an ability a few generations later that makes its Bullet Punch one of the most feared attacks in the game, even in Ubers. Epic is the only way to describe Scizor.
Forretress. Created specifically to Stone Wall the normal-type kings from Generation I, and later becomes one of the best Pokémon to set up entry hazards.
Politoed. Long forgotten as that random green toad from Johto, the moment people realized the thing will be given Kyogre's Drizzle ability, people went nuts to have it added to their Rain teams.
Tyranitar, Generation II's pseudo-legendary. It helps being one of the only pseudo-legendaries, apart from Metagross, that's NOT a Dragon type. Gaining Sand Stream to summon instant Sandstorms certainly helped his cause in competitive play.
Pokémon - Generation III
Ludicolo has been one ever since it was solidified as Miror B's (a darkhorse in his own right) signature Pokémon. Its Grass/Water typing is also quite a godsend in competitive play; having weaknesses to just Poison, Bug and Flying, which, other then Flying, are very uncommon types for attacking.
Aggron, who restores mountains and looks like a Badass metal version of another darkhorse, Tyranitar.
Flygon; its unique typing and design stood out amongst all other Dragon Pokemon...at least before the Gible line was created. This, however, did not diminish its popularity; a dedicated and strong fan following still prefers to use it over Garchomp. It's still useful as a scout with resistance to Earthquake (something Garchomp does not have) and entry hazards like spikes and Toxic spikes, and its design is still considered quite unique.
Groudon and Kyogre, the mascots of Ruby and Sapphire whose weather-starting abilities would go on to redefine the metagame. Kyogre is known as the "King of Ubers" on Smogon, and for good reason.
Pokémon - Generation IV
The starting point of this Generation should be none other then it's unofficial mascot, Lucario. His preevolution, Riolu is popular as well.
Starly, a Com Mon similar to Pidgey who quickly becomes a Disc One Nuke. It learns great moves like Wing Attack early on, and eventually evolves into the Big Badass Bird of Prey Staraptor, one of the rare Flying types capable of learning Close Combat. In fact, many have considered it one of the best Pokémon for completing the main story, period.
Glaceon. It's like an Ice-type Espeon, except that it's a LITERAL Mighty Glacier.
Its counterpart, the long awaited Leafeon, isn't that far behind in popularity.
Hippowdon use caught on post-Generation IV with the many additions made in Generation V; increasing the utility of using Sand teams to take advantage of Hippowdon's ability to summon instant Sandstorms. Unlike Tyranitar (who also gets Sand Stream), it has Slack Off, much more physical bulk, and FAR fewer weaknesses (3 as opposed to Tyranitar's 7), making it a very good wall.
Abomasnow is often overlooked, but has its own loyal fanbase following amongst competitive players; particularly for being the staple of Hail teams as the only Pokémon that can summon instant Hail. Its typing sure doesn't do it any favors; having seven weaknesses turns many players away at first glance. However, it turns out that the weaknesses consist primarily of the lesser-used offensive types in competitive play, such as Poison, Steel, and Bug, and that the resistances Abomasnow does have are actually quite useful in competitive play; Water, Electric, and Ground specifically. The fact Abomasnow was one of the few Generation IV Pokémon to receive a Mega Evolution is no coincidence. On a side note, it's also quite prone to Barack Obamasnow jokes.
Garchomp, Generation IV's pseudo-legendary. Particularly popular with the older crowd because of its incredible power, and popular with the younger crowd because it resembles what you'd get if you combined a Dragon, a shark, and a jet-engine fighter plane.
Darkrai is immensely popular for its design, battle style, and two WIDELY different depictions (The Movie Darkrai and the Mystery Dungeon Darkrai have virtually no similarities).
Scrafty is also a very well-liked Pokémon from Generation V. It looks cool, has an interesting theme, and is uniquely useful in competitive play; it and Scraggy (its previous form) are the only dark/fighting types, which means they're the only fighting type Pokémon immune to psychic attacks note Well, at least before Generation VI's Pangoro came along, which is fighting/dark and, therefore, shares similar immunities and weaknesses, and they get STAB on two very useful types. Scrafty's possible abilities are also very good. Moxie increases its attack every time it defeats an enemy, making it a good sweeper. Shed Skin has a chance of removing a status effect, so it's great for a defensive set. The official Nintendo 2011 Black and White tournament gave out a fairly decent Scrafty as a nod to its popularity as a team fighter. It likely helps that Scrafty is really useful in-game as well, with its typing rendering it good against three of the Elite Four.
Chandelure. Right up there with Gengar as one of the best Ghost types of the game thanks to its god-like special-attack and amazing type coverage. Which is ironic considering the status most other Pokémon based on inanimate objects get.
Golurk is a freaking Humongous Mecha! The moment people figure out it can learn Fly ups anyone's view of the giant robot. Also has the unique typing of Ghost/Ground.
Hydreigon, Generation V's pseudo-legendary and the signature 'mon of the aforementioned Ghetsis.
Genesect, a prehistoric insect-turned-cyborg weapon of mass destruction. Based on the trailer for the next movie, a Transformer too. This only served to increase its popularity.
Zekrom is easily the most popular Unova Pokémon (rather than Lucario's own Expy, Zoroark). Especially in America; in the aforementioned IGN popularity poll, despite Black and White being completely new in the States, Zekrom managed to break the Top 10.
Though both Mega Evolutions of Charizard are popular, Mega Charizard X clearly is the more popular of the two, mostly for giving Charizard the secondary Dragon typing fans have been clamoring for. It's often believed to be at least partly responsible for X selling more copies than Y despite Y having the more popular version mascot. Its black-and-blue color scheme also helps. Not to mention that both Mega Evolutions are a huge help to Charizard in the metagame, turning it from an unusable Tier-Induced Scrappy to a serious threat in either form.
Skiddo and Gogoat. Both can be ridden on in game. Skiddo is a cute little goat kid, while Gogoat is cross between an adult goat and a motorcycle with good stats and a surprisingly large movepool when compared to other Grass-types.
Honedge and its evolutions, the possessed ghost swords. Especially the final evolution Aegislash, with an Awesome Mc Cool Name in every language, a cool semi-humanoid design, and great viability in competitive play.
Tyrunt and Tyrantrum, the first trueTyrannosaurus rex Mons in the series - not just by virtue of being clearly based on a popular dinosaur known for being supremely Badass, but also by virtue of being Dragon-types!
The sauropod Ice fossils Amaura and Aurorus didn't go unappreciated, either. You'd be lucky to come across a Pokemon forum that hasn't made a The Land Before Time or Jurassic Park joke with the new Poké-dinos.
Pumpkaboo and Gourgeist, one of two new Ghost/Grass type families introduced this generation clearly saw more popularity than the Phantump/Trevenant line for their adorable, unique, and creative design (Halloween and pumpkins) and for coming in four different and unique sizes ranging from small to super size which isn't simply a cosmetic difference, but also are seperated from one another from how one uses them (small tends to be speedy sub-seeders whilst the Super-Size are very bulky).
Conway, an intended one-shot who was added to two other story arcs (the Pokémon Summer Academy and the Sinnoh League) due to fan popularity.
Some of the Characters of the Day get this as well. Gligarman (an obvious Batman and superhero parody) was popular enough to frequently get requested to return to the show, especially when Ash had a Gligar/Gliscor of his own.
Harley, because he's such a Jerkass and yet sodamnFUNNY at the same time. He was also the first genuinely antagonistic rival in the series who remained an antagonist and not change his behavior, which the fans greatly appreciated.
Ursula, a one-shot character of the day that becomes a full-fledged rival for Dawn in the next season due to fan popularity for being an antagonistic rival; separating her from Zoey, Kenny, and Nando.
Amongst Ash's numerous rivals, Stephan has become one of these. Being less prominent than Trip or Bianca (among others), he nonetheless has a lively and likeable personality, as well as being a Badass battler from what we get to see of his skills. It also helps that he gets what is, hands down, the best battle in the entire Unova League.
The Club Tournament contestants - many don't even get any lines, but their character designs are regarded as among the best parts of those arcs (Antonio and Jimmy Ray/Watchog Guy especially). Some of these characters, like the aforementioned Antonio, return for the Unova League.
Edmund the "Rain Man", who's an (almost literal) One-Scene Wonder (he appears momentarily before battling Stephan, and disappears altogether afterwards).
Despite only appearing in one game and a single episode, Lucy the Frontier Brain is very popular and one of the main characters shipped with Brock.
Similar to the above Lucy example, several girls are this for shippings with Ash despite not being one of his primary traveling companions.
Bianca from Pokémon Heroes for, again, cheek kissing Ash at the end of the movie. (Though, if this was actually Latias has always been up for debate)
Frontier Brain, Anabel, maintains a fanbase for Ability Shipping.
Angie, a tomboy from the Sinnoh Summer Academy arc for Morpheus Shipping.
Several of Ash's Pokémon have developed rather large Fandoms over the course of the series due to their sheer Badassery or quirky personalities. The most notable being Charizard, Heracross, Swellow, Sceptile, and Staraptor. Even Ash's unofficial Pokémon, Haunter and Larvitar, have a place in fans' hearts. Also, the anime made Pikachu one. It was just another Pokémon before that.
Psyduck and Wobbuffet, two very similar, dopey Pokémon who just won't stay in their Poké Balls.
The Ducklett trio from an early Best Wishes episode, who caused bizarre, yet adorable chaos and trolled the characters several times. Ask any Pokémon forum about them, and you'll see they're almost universally loved.