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, since Generation IV, it is possible to collect foreign Pokédex entries, and there are NPCs who 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 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 and their Generation VI remakes, 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 is 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. Also, in Pokéstar Studios, the final film of the Love and Battles Series has a talking Smeargle, and the Timegate Traveler Series features a talking Ledian.
In Sun and Moon and Ultra Sun and Ultra Moon, the most notable Pokémon that speaks is Rotom. However, Rotom is only able to speak when inside the Rotom Pokédex. Tapu Koko also speaks to the player through their Z-Ring before challenging them to a battle at the end of the game. A Mimikyu at the Aether House tells the player it will curse them when spoken to as well. An Oranguru in the Konikoni City apparel shop also tells the player to "Gu do it!"
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's Pokédex entry, where it describes its ability of human speech comprehension. The Pokédex entries for Latias state that it is able to understand human speech, and it is also said that it can telepathically speak with humans.
In the Mystery Dungeon games and special anime episodes, all Pokémon can talk (minus Jynx). 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 Detective Pikachu, the player character, Tim Goodman, is partners with a talking Detective Pikachu. However, only the player seems to be able to hear Pikachu's talking voice. In addition, the game also features a Mewtwo, which is able to talk via telepathy.
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 Larvitar 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 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 and Mythical Pokémon can. In most cases though, talking Pokémon speak through telepathy.
Meowth is the most famous talking Pokémon, and the one that makes the most appearances. His clone in Mewtwo Strikes Back and Mewtwo Returns, however, cannot speak, the reason being Meowth wasn't born knowing how to speak human language: he only learned it through hard work.
Although Ash's Pikachu does not speak in the regular anime, he has talked once in Pokémon the Movie: I Choose You!, when Ash seemed to die from taking the attack from various Pokémon late in the movie.
In the manga
In the Magical Pokémon Journey manga
Clefairy and Squirtle can speak human language, although most Pokémon 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 Adventures manga
In ...But Fearow Itself!, when Bill was accidentally fused with a Rattata, he was still able to talk and ask Red to aid him in reversing the transformation.
In Red, Green, Blue and Mewtwo Too, Mewtwo revealed to Red that it had gained the ability to talk via telepathy after leaving Blaine in Heckled by Hitmontop, although it usually only lets its voice be heard by people whom it trusts.
In Gotcha Where I Wantcha, Glalie, it was revealed that Latios and Latias are also able to speak via telepathy.
In The Shadow Triad, the Swords of Justice and Keldeo were revealed to be able to speak telepathically, although it was later revealed that humans can only hear this speech under certain conditions, such as by being at the Abyssal Ruins.
In Quilladin Stands, Xerneas was shown to be able to use telepathy to communicate with humans as well.
In PASM07, the Rotom that Moon had brought with her to Alola was revealed to have entered her new Pokédex and become the Rotom Pokédex. Through the Pokédex, Rotom had gained the ability to speak human language.
In the Pokémon Gold & Silver: The Golden Boys manga
In The Truth About The Mysterious Whirl Islands!!, Gold and Chris encountered a talking Slowking living at the Whirl Islands.
In the Pokémon Gotta Catch 'Em All manga
Shu can understand his Pokémon's language by using a special headset device called an Income.
In the Pokémon Pocket Monsters manga
All the Pokémon in the Pokémon Pocket Monsters manga can talk, except for Red's Pikachu.
In the Pokémon Zensho manga
In Vermilion City, Satoshi encountered and caught a talking Nidoran♂, who soon turned out to be Bill, who had accidentally gotten himself fused with a Nidoran♂ during an experiment. Afterwards, Satoshi helped Bill to become normal again.
In Saffron City, Sabrina was revealed to own a unique telepathic Lapras that could communicate with humans. After Satoshi had helped her save Lapras from Team Rocket, she entrusted Lapras to him as a sign of gratitude.