If you want a basic overview of the biggest ships in the fandom, they go something like this:
PokéShipping: Girl catches boy on her fishing line. Boy steals her bike. Boy's electric rat fries girl's bike fighting Spearow. Girl get mad at boy. Girl follows boy around, ostensibly to remind him he owes her a bike. Boy and girl become friends. Fandom is never the same again.
AdvanceShipping: Girl starts journey. Boy's Pikachu has gone feral. Boy and girl wind up foiling Team Rocket's plan, frying girl's bike. Girl ropes boy into traveling with her. Fandom is never the same again.
PearlShipping: Girl finds boy's amnesiac Pikachu. Boy and boy's friend meet up with her and Pikachu, foil Team Rocket scheme. Girl's bike is fried. Boy, girl, and friend decide to travel together; girl acts as boy's cheerleader in Gym battles. At this point fandom saw this coming and is relatively unchanged.
Ash/Misty, Ash/May, Ash/Dawn. PokeShipping, AdvanceShipping, PearlShipping. The "big three" Shippings around, to be sure (soon to be a "big four" once NegaiShipping (Ash/Iris) gains more prominence). Along with RocketShipping (Jessie/James), PalletShipping (Ash/Gary), and ContestShipping (May/Drew), these are the most widely-supported Ships in the whole of the Pokemon fandom - and, since they all involve the main character and his female traveling companions, the biggest intership rivals.
PokéShipping is one of the original Pokemon Shippings. Sometimes called Ash And Misty Love, or AAML, there is some evidence that it was originally intended to become canon - though the late Takeshi Shudo denied this. PokeShipping was one side of the infamous Shipping wars of the 90s, with current Bulbagarden owner Archaic as one of its supporters. Its only really big rival at the time was PalletShipping. PokéShipping still maintains a devoted following to this day. Contrary to popular belief, PokéShipping is not a product of the Pokemon dub (although this perception is hardly helped by dub material such as "Misty's Song"), and indeed many PokeShippers only count the original Japanese as canon when looking for PokeShipping "hints."
PokéShipping seems plausible enough from a realistic standpoint. Opposites attract, as the saying goes. As mentioned before, the Ship may have even been intended to be canon at some point, though this clearly is no longer the case.
AdvanceShipping, the pairing of Ash and May, was a somewhat unexpected addition to the Pokemon fandom, as the addition of May herself was. Well, perhaps not unexpected, but at the time unique. In AdvanceShipping, PokeShipping found its first serious rival since PalletShipping in the 90s. The relationship between Ash and May in canon is a charming teacher-student one, and it does not seem too far-fetched (or, since it is obligatory to make this joke in a Pokemon article, Farfetch'd) for two pleasant and friendly individuals to wind up as a couple.
In fact, if one believes a certain Pokedex entry, AdvanceShipping is fated to be. In the ninth movie, Ash and May are diving in the sea when a school of Luvdisc surround them. According to the Pokemon's Dex entry, a couple who sees a school of Luvdisc is fated to be together forever. The Shippiness of the scene is dependant on many factors, including taking Pokédex entries at face value, one's definition of what constitutes a couple, and even whether or not the films are canon.
PearlShipping, or Ash/Dawn, is one of the more recent Ships to gain prominence in the fandom. Each Ship has its symbols - PokeShipping has the lure that Misty gave to Ash, and AdvanceShipping has the ribbon Ash and May split between each other - and if a symbol had to be chosen for PearlShipping, it is likely that the symbol would be a high five. Supporters of PearlShipping are quite fond of the high fives that Ash and Dawn often shared, and these high fives are one of the Ship's most oft-mentioned hints.
PearlShippers also hold the anime opening "High Touch" in high regard - the opening was sung by Ash and Dawn's Japanese voice actors, and contains several lines that can be construed as Shippy. The fact that Dawn would often act as Ash's cheerleader during Gym matches is another hint that PearlShippers enjoy bringing up.
The anime is constantly evolving, and as more characters are added to its cast, more and more Ships will rise to popularity. For now, though, the Ships that have the largest following stand above the rest - PokeShipping, AdvanceShipping, and PearlShipping. The Big Three. And, with the recent addition of Iris to the show's cast, this will evolve into a Big Four. And then, a Big Five, and so on, for as long as the show airs.
And the Shippers? They wouldn't have it any other way.