Since the 1.1 update I've retrieved a few statues from underground houses. While I'm pretty sure that the bunny statue spawns bunnies, and the slime statue spawns slimes, I have no idea what the rest of them do.
According to the wiki there are three types of functional statue:
The item spawning statues can spawn an item approximately every ten seconds.
There is a limit of three spawned monsters of a given type that can exist in the world at one time. The monsters spawned from statues do not drop money, but, with exception of the mimic, do drop items. During a bloodmoon bunnies and goldfish will immediately turn corrupt and will drop money when killed. Because the corrupt versions do not count towards the 3 spawn limit, they can be spawned and harvested for cash quickly.
Any statues not listed do not have a known function.
They do various things. Some will spawn monsters. It is speculated that they need to be hooked up to a source of power via a wire. The exact mechanics of this are still being investigated, I believe.
From the version 1.1 changelog:
When information is available, it will be on the Terraria Wiki, and I'll try to come back and edit this to give a better reference.