PokéShipping

Pokéshipping, also well known as SatoKasu (サトカス) in Japanese, is the het ship between Ash and Misty in the anime series of Pokémon.

Canon
Ash and Misty met in the first episode of the anime, when Misty fished Ash and Pikachu out of a river. Ash and his Pokémon were running from a flock of Sparrow at the time, so he 'borrowed' Misty's bike to get Pikachu to a Pokémon Center as quickly as possible. The wild chase inevitably ended with Ash and Pikachu completely destroying her bike, and as a result, Misty made it her mission to track him down and get him to pay her back. However, after tagging along with Ash for a period of time, Misty grew attached to him, eventually caring about him enough to call him her best friend and even develop an often hidden crush. The bike was eventually forgotten completely, and after all the adventures, trials, and life changing moments they experienced on their journey together, the two consider each other very good friends to this day.

During their time traveling together, others often referred to Ash and Misty as though they were a couple, which sparked embarrassed blushes and denial from both of them almost every single time, without fail. Some notable in-canon shippers include Tracey, Daisy, Nurse Joy, Melody, Danny, and even Team Rocket. Ash and Misty both had a habit of getting jealous when a potential suitor showed interest in the other. Notable examples of this include Misty's anger at Giselle when she managed to get a blush out of Ash, Misty's shocked and angered reactions when Melody and Bianca kissed Ash in the second and fifth movie respectively, Misty's rivalry with Macey, which was spurred on by Macey's open flirting with Ash, and Ash's angry reaction toward Fiorella, Danny, and Rudy managing to capture Misty's romantic interest. The last three examples are particularly notable because they are the only time in the anime's excessively long run that Ash has ever shown jealousy over a girl. There are also a large number of other affectionate displays between the two, some of which may be left up to interpretation and some of which are objective signs of romance, but in the interest of not flooding this page, you may find two fairly thorough lists here and here.

Ash and Misty are no longer traveling together in the current iteration of the anime, but references have been made to the fact that they still keep in touch. Misty is seen watching several of Ash's big battles, Misty appears to be already filled in on Ash's current adventures every time she meets up with him, and the Best Wishes ending, Let's Join Hands,out and out confirms that Ash still keeps in contact with his former traveling companions via e-mail, letters, and phone calls. Misty was revisited in the Pokémon special, A Date With Delcatty, in which it is heavily implied that she still has strong feelings for Ash and is waiting for him to return from his journey. When a boy named Georgio asks her out on a date, she tells him she can't go, to which her friend Casey asks her if there is someone else she has in mind. Misty blushes and nervously avoids the question only for her Psyduck to come out of its pokéball and chuckle at her because it knows who she really likes. Ash's feelings for Misty were hinted at again in the Diamond and Pearl episode, Buizel Your Way Out of This!, in which Ash is shown to be excessively protective of the special fishing lure Misty gave him as a present, going as far as to keep it away from his friend Dawn and practically fight her for it when she wants to use it. Later in the episode, a wild Buizel steals the lure and the rod its attached to for itself, causing Ash to panic, become angry, and devote a significant portion of the episode to getting it back. When he finally does retrieve it, he is noticeably relieved.

Fanon
Pokéshipping is a very popular Pokémon ship. it rivals all other Ash and Misty pairings, including, but not limited to Palletshipping (Ash x Gary), Advanceshipping (Ash x May), Pearlshipping (Ash x Dawn), Negaishipping (Ash x Iris), Amourshipping (Ash x Serena), Altoshipping (Ash x Latias), Ash x Lillie (S&M), Ash x Lana (S&M), Ash x Mallow(S&M, Egoshipping (Misty x Gary), and Orangeshipping (Misty x Tracey).

Trivia

 * Thus far, Misty is the only one of Ash's traveling companions to ever get a strong reaction out of him on the subject of romance. She is the only traveling companion he has ever given a crush blush to and the only traveling companion he has ever gotten jealous over.
 * There is a long history of ship wars in the Pokémon fandom. Pokéshipping, being one of the oldest ships, has been prevalent throughout pretty much every era of ship-to-ship combat. It participated in a massive war with the Palletshipping (Ash x Gary) fanbase near the beginning of the series, there was a war between Pokéshipping and Advanceshipping (Ash x May) during the Advanced Generation, and Pokéshipping and Pearlshipping (Ash x Dawn) fought in the DP era. Right now, Pokéshipping and Amourshipping (Ash x Serena) are the two largest ships in the fandom, and the ship wars are still continuing on.
 * Misty has romantic feelings toward the "Red character" (Ash variants) in multiple interpretations of the Pokémon story. In the Pokémon Adventures manga, Misty is shown to have a crush on Red early on and even struggles to confess to him in later chapters. In the The Electric Tale of Pikachu manga, Ash and Misty are again shown to have crushes on one another and even openly flirt from time to time. Misty's crush is further referenced in Pokémon Comics, the Ash & Pikachu manga, Pokémon Live!, and various 4Kids spin off songs and promotional material.
 * Pokéshipping has also received a wide variety of references outside of the official Pokémon franchise throughout pop culture. The romantic undertones of Ash and Misty's relationship were hinted at in episodes of Robot Chicken, MAD, Pokémon 'Bridged, Smosh, Game Theory, Maré Odomo's comic series, Letters to an Absent Father, the StarBomb song, I Choose You To Die, and the ALL CAPS song, Trainers in Love. An unofficial Pokémon musical, appropriately titled, Pokémon: The Mew-sical was also put on by a handful of students at Yale University with one of the major plot points being Ash and Misty's difficulty confessing their feelings to one another. A recording of the show can be viewed in its entirety here.