From Bulbapedia, the community-driven Pokémon encyclopedia.
Talking Pokémon are Pokémon who can speak a human language. The human language itself is not clearly depicted and all people in the Pokémon world seem to speak one common language, which is always referred to just as human language (however, in Generation IV games it is possible to collect foreign Pokédex entries, and there are NPCs in Generation IV and Generation V that speak foreign languages). While it appears that all Pokémon can understand human speech, very few can communicate with humans except via body language. There are exceptions to this, however, depending on the version of the Pokémon world being depicted.
In the games
In Red, Blue, and Yellow Versions and their Generation III remakes, Copycat's Doduo is able to speak a line of human language when approached. Also, Bill, when transformed into a Pokémon, can speak human language from his house on Cerulean Cape. In addition, the Marowak's ghost can talk.
In Generation II and their Generation IV remakes, a Murkrow owned by Team Rocket tells the player the password to the generator room in Mahogany Town.
In Ruby, Sapphire, and Emerald Versions, Mr. Stone wants to be able to communicate with Pokémon, so he started a project for Devon Corporation intending to build a translation device for Pokémon; however, not much progress has been made. Dr. Kaminko developed a device prior to XD that was meant to accomplish this, but it failed and was scrapped. A major post-ending quest details the player's efforts to undo its effects when it is accidentally released by Chobin.
Generation IV introduced Chatot, a Pokémon mentioned as being able to speak the human language through the use of its Chatter move.
Darkrai is not able to speak directly, but it's presumed that the man inside the Harbor Inn is an illusion created by Darkrai, through which it is able to communicate with the player.
In Black 2 and White 2, if the player visits Lostlorn Forest, a Zoroark is disguised as a backpacker who talks to the player before dispelling its illusion and leaving.
In Hey You, Pikachu!, the game centers around using a special device called a PokéHelper that translates certain human-language words into something wild Pikachu can understand. In this game's spiritual successor, Pokémon Channel, various Pokémon appear as hosts and stars of TV shows, with their speech subtitled into the local human language.
Certain Pokémon are also capable of using telepathy, such as the Regigigas in Pokémon Ranger: Shadows of Almia.
It may not be known if all Pokémon, wild or not, are able to understand everyday human speech in the games, as said in Lapras' Pokédex entry, where it describes its ability of human speech comprehension. Latias also is said in its Pokédex entries that it is able to understand Human speech and is said it can also telepathically speak with Humans.
In the Mystery Dungeon games and special anime episodes, all Pokémon can talk. This includes the player, who was transformed into a Pokémon. However, since there are no human characters in the Mystery Dungeon games, it is entirely possible that these Pokémon are not, in fact, speaking human language, but rather their own Pokémon language, which the player can understand as a result of being a Pokémon himself/herself.
In the anime
Understanding of human language
As mentioned above, all Pokémon in the Pokémon anime, with no exceptions, can understand human speech. Moreover, judging, for instance, from Phanpy in Hatch Me If You Can, they are able to do this immediately after hatching, without having to learn.
The anime generally does not focus on language problems, so as a result there is some confusion about Pokémon language and the ability of Pokémon to understand humans. Although it is obvious that all Pokémon can understand each other, Meowth in the episode Go West Young Meowth once mentions Meowth language and Jessie in Who's Flying Now? mentions Chimecho language, so it is not clear whether there is a common Pokémon language or every species has its own.
As far as human language is concerned, Meowth in the same episode was shown learning pronunciation, not the language itself, what implies that all Pokémon have innate knowledge of human language, so they can understand human speech and only have to learn how to pronounce its sounds or be able to establish telepathic contacts to speak. Despite this, Misty in Whichever Way the Wind Blows once asked Meowth to translate her words to other Pokémon and May taught her Bulbasaur the meaning of the word town in Grass Hysteria. Therefore, this ability of Pokémon is not well-defined in anime canon.
Although every Pokémon in the anime can understand human speech, very few can talk, although many legendary Pokémon can. In most cases though, talking Pokémon speak through telepathy.
In the manga
In the Pokémon Pocket Monsters manga, all the Pokémon can talk except for Red's Pikachu.
In the Magical Pokémon Journey manga, Clefairy and Squirtle can speak human language, however most Poké-speak is translated for the reader's ease (indicated by a different font in the word bubbles). They are easily understood by humans just the same.
In the Pokémon Get da ze! manga, Shū can understand his Pokémon's language by using a special headset device called an In-Com.
In the Pokémon Zensho manga, Sabrina has a unique telepathic Lapras that can communicate with humans.
In the Pokémon Adventures manga, when Bill is accidentally transformed into a mutated Rattata in ...But Fearow Itself! he asks Red to aid him from the transformation.