From Bulbapedia, the community-driven Pokémon encyclopedia.
Talking Pokémon are Pokémon who can speak human language (what that language is depends on your particular location). 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:
Talking Pokémon in video games
In the Pokémon games, Mr. Stone wants to be able to communicate with Pokémon, so he's having Devon Corporation build a translation device for Pokémon. However, the development isn't going so well.
Chatot is a Pokémon mentioned as being able to able to speak the human language through use of its Chatter move.
Copycat's Doduo is able to speak a line of human language when approached.
A Murkrow owned by Team Rocket in G/S/C tells the player the password to the generator room in Mahogany Town
Talking Pokémon in the anime
In the Pokémon anime, very few Pokémon can talk, although many legendary Pokémon can.
Talking Pokémon in manga
In the Pokémon PiPiPi Adventure 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 Getto Da Ze! manga, Shū can understand his Pokémon 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 Special manga
When Bill is accidentally transformed into a mutated Rattata in ...But Fearow Itself! he talks to Red to help him.