Birds that talk include the African Grey Parrot, who not only talks, but also takes on the various tones of the people who talk to him and can use several different voices. The various Amazon Parrots talk (some better than others) and are also known for their singing voices. The little Budgies often have a large vocabulary, but can be difficult to understand until you have become accustomed to their voices.
Other birds in the parrot species that are known for their talking ability are Cockatiels, Eclectus, some of the Cockatoos, some of the Conures and Macaws. Birds such as Lovebirds can occasionally also talk. With Cockatiels & Budgies, you will find that normally only the males will talk, but with Cockatiels especially, you will find that the females are often more cuddly and make marvelous pets.
Mynahs and Crows are not Parrots, but are also considered talking birds.
Click here to hear my African GreyParrot Jewel say "Aren't you adorable".
See How to Teach Your Parrot to Talk for help in teaching your own parrot to talk, including the story of teaching Jewel how to say "Aren't you adorable".