What do the new statues do?

According to the wiki there are three types of functional statue:

1. Statues that spawn items
   • Bomb statue - spawns bombs
   • Heart statue - spawns hearts
   • Star statue - spawns mana stars
2. Statues that spawn enemies
   • Bat statue -spawns a bat
   • Bird statue - spawns a bird
   • Bunny statue - spawns a delicious dinner (or a murderous beast during a bloodmoon)
   • Chest statue - spawns a mimic
   • Crab statue - spawns a crab
   • Fish statue - spawns a goldfish (corrupted during a bloodmoon)
   • Jellyfish statue - spawns a jellyfish
   • Piranha - spawns a piranha
   • Skeleton - spawns a skeleton
   • Slime - spawns a slime
3. Statues that teleport NPCs
   • King statue - teleports a random male NPC to the statue
   • Queen statue - teleports a random female NPC to the statue
4. Bonus: Dart trap - fires a dart which damages and has a chance to poison the creature hit

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:

Added thirty-nine statues that are scattered around the world. Some of them do special things when activated with mechanics!

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.