In the anime
>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 learned how to speak human language.
In Island of the Giant Pokémon, Ash's and Team Rocket's Pokémon can speak due to most of the episode focusing at them (although this is done through subtitles in the Japanese version and several of the dub versions). This prevents the difficulty of understanding Pokémon saying only the syllables of their names.
A Tentacruel in the episode Tentacool and Tentacruel speaks through Meowth via telepathy.
A Gastly in The Ghost of Maiden's Peak has the ability to talk.
Santa Claus's Lapras in Holiday Hi-Jynx has the ability to talk telepathically.
Both Mewtwo seen in the anime can talk via telepathy.
The Gastly and Haunter in the episode A Shipful of Shivers speak through Meowth via telepathy.
The Lugia in The Power of One also speaks via telepathy.
A Slowking in The Power of One can also talk without telepathy. Presumably this is because of its wisdom.
Entei can talk in Spell of the Unown. However, this Pokémon is the embodiment of Molly's wishes, so therefore it is not currently known if Entei itself can speak.
Ash's Larvitar in Address Unown was speaking when Ash and his friends entered its mind.
Max's Ralts in Do I Hear a Ralts? talked to Max as it teleported away via telepathy.
Deoxys from Pokémon Ranger - Deoxys' Crisis! (Part 1) and (Part 2) spoke through Meowth telepathically.
Jirachi can talk via telepathy in Jirachi: Wish Maker.
Lucario can talk in Lucario and the Mystery of Mew via telepathy.
Manaphy can talk without using telepathy in Pokémon Ranger and the Temple of the Sea, although it can only say a few words (such as "happy" and "I love you") as it is still a baby.
Chatot can repeat human words.
Darkrai occasionally speaks in The Rise of Darkrai.
Shaymin can speak telepathically in Giratina and the Sky Warrior.
Arceus is able to speak human language in Arceus and the Jewel of Life.
A Snover from the Ice Palace can mysteriously talk in Pikachu's Ice Adventure.
Zorua can speak telepathically in Zoroark: Master of Illusions.
A Yamask speaks through Cilan in A Night in the Nacrene City Museum! while possessing him with its mask.
Zekrom and Reshiram are able to speak telepathically in White—Victini and Zekrom and Black—Victini and Reshiram.
Kyurem and the Swords of Justice, including Keldeo, are able to speak via telepathy in Kyurem VS. The Sword of Justice.
The members of the Genesect Army in Genesect and the Legend Awakened are able to speak via telepathy.
In The Island of Illusions!, a Zoroark is able to talk by changing its form into a Nurse Joy.
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 Gotta Catch 'Em All manga, Shu can understand his Pokémon's language by using a special headset device called an Income.
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. In the FireRed & LeafGreen chapter, Mewtwo reveals to Red that he gained the ability to talk via telepathy after leaving Blaine.
In the video 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 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 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 (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.
Hope I helped! :D