It is clear that Harry can speak to snakes and Dumbledore can understand some parseltongue too:
The ability to speak Parseltongue is extremely rare, and is something for which Salazar Slytherin was famous (hence his nickname of "Serpent-tongue", and the choice of a serpent as the symbol of Slytherin House). Tom Riddle inherited this ability from Slytherin, and used it to control the basilisk from the Chamber of Secrets. Harry acquired the ability to speak Parseltongue from Voldemort during Voldemort's attack on the Potters. Harry used the ability unwittingly to speak to a boa constrictor in the zoo and to a snake conjured by Draco Malfoy at the Duelling Club.
Parseltongue is the language of snakes. It is in the common mind associated with Dark Magic (although Albus Dumbledore himself has denied it to be in any way evil), and those possessing the ability to speak it ("parselmouths") are very rare. It appears to be a skill acquired not through learning, but through genetic inheritance (or by use of Dark or dangerous Magic). Harry Potter is a Parselmouth: it is speculated at various points in the series that his acquisition of this skill was one of the many by-products of Lord Voldemort's attempt on his life when he was a child.
Other known parselmouths include: Salazar Slytherin and his descendants, including the Gaunts (Marvolo, Morfin, Merope), Lord Voldemort, and Herpo the Foul. Albus Dumbledore seems to be able to understand Parseltongue in one instance in Harry Potter and the Half-Blood Prince, when he repeats Morfin Gaunt's words "the big house over the way", which were spoken in Parseltongue. This means that either he himself is a Parselmouth or that he had previously viewed that memory with a Parselmouth who translated the conversation for him. It is unlikely to have been Morfin who did so: Dumbledore claims that, "apparently, ["He'll kill me for losing his ring"] was all he ever said again."
Ginny Weasley is not a parselmouth, despite the fact that at points in Chamber of Secrets she appeared to speak in parseltongue: she was at the time being possessed by the spirit of Tom Riddle, who was himself a parselmouth.
