Walking Pokémon

Walking Pokémon are Pokémon that walk alongside their Trainer when they're normally would be inside the Poke Ball.

Games
Pikachu was the first Pokémon that followed the trainer around and the player character could interact with it. This feature was added in Pokémon Yellow Version: Special Pikachu Edition. Pikachu could not follow the trainer if it was lulled to sleep or it fainted.

Anime
The most obvious example is Ash's Pikachu. During Ash's first day as a Pokémon Trainer, Pikachu adamantly refuse to be inside the Poké Ball after being sent out by Professor Oak, and afterward revealed to Ash that he didn't like being inside the ball, thus Pikachu often rides on Ash's shoulder or head.

Misty's Togepi is another Pokémon in the anime famous for never being recalled to a Poké Ball, and even more notably, having never been seen to be in a Poké Ball at all. Ash found it as an Egg in Grampa Canyon in Attack of the Prehistoric Pokémon, and it was cared for by Brock until it was stolen by Team Rocket, subsequently hatching in Misty's arms. It served as a companion to Pikachu throughout the series from its debut until Misty's departure, and was eventually released by Misty in A Togepi Mirage!. Some do not count it as a walking Pokémon due to the fact that it never had any Poké Ball, even though it acted like a caught Pokémon.

Like Pikachu, Dawn's Piplup has also become a walking Pokémon, though he was initially completely fine with staying inside his Poke Ball at the beginning of the Diamond & Pearl series, and has not indicated that he has ever had a problem with it, unlike Pikachu. Since Journey To The Unown!, Dawn often carries him around in her arms much like Misty did Togepi. In some episodes, however, Piplup instead rides on either her head or her shoulder.

Another example is Iris' Axew, as instead of staying inside the Poké Ball he enjoys riding within Iris' hair. It is unknown if Axew has any dislike of a Poké Ball, as the issue of if he even has one has not been raised in the series and also because when the elder first presented him to Iris he was seen in a basket without a Poké Ball around it.

Manga
In Yellow arc, Yellow has Red's Pika follow her from "YL002: Do do that Doduo". Pika's Poké Ball was in Red's possession and both Pika and Yellow went to search for Red, who went missing.

Trivia

 * Most of the recurring walking Pokémon in the anime have not evolved or refuse to do so. The only exceptions are Misty's Togetic, Cameron's Lucario, and Sanpei's Greninja, although Togetic was released soon after evolving. Iris' Axew's dream in "Dreams by the Yard Full!" revealed that he wants to evolve into his final form.
 * Advanced Generation is the only series to date not to have any walking Pokémon besides Pikachu in Ash's regular group.