How do I sleep or skip nights?

I was able to build a bed in Terraria and I thought it could allow me to sleep. By clicking on it, instead, I set only the spawn point.

Is it possible sleep in other ways or eventually skip nights (or days)?

The 1.3 version of Terraria has now a solution for you.

This item is the Enchanted Sundial.
It is a furniture that can skip one day/night cycle, by setting the time at 4:30 AM. It will also create a new Angler quest.
It reloads every week of in-game time, and no legit action can hasten the cooldown of the furniture :

-placing multiple sundials ?
-remove and place the sundial again ?
Nope, the sundial's state is stored in the world's data, you'll have to wait a full week of in game time, which is 168 minutes (= 2h48m) before the item gets usable again.

It can be found inside Wooden, Iron or Golden crates, opened in a hardmode world, with better odds of getting it respectively.

This is not currently possible in the 1.01 version of Terraria, beds are strictly for setting of your spawn point.

UPDATE: I think this is currently a bug, but in 1.0.3 (And maybe before) if you play in windowed mode, minimize the game by clicking on its icon on the task bar (NOT using the minimize button on the window itself) the game runs VERY fast. You can still hear the background music and wait for it to play the day music... Takes about 15 seconds or so. Also note that the days will go by fast when you do this AND you need to be somewhere 100% safe as, well the game is still running and things will still try to get you.

I think the game is using a fixed time step because of its fixed framerate, but for some reason the framerate jumps up beyond the normal 60 max when minimized in this fashion and so runs very fast.

If you are running the server console (TerrariaServer.exe, should be in the folder you installed to) yourself, you can enter into the console the commands "dusk", "midnight", "dawn", or "noon" to change the time of day to that time. Even when playing single player, I tend to run the server console and connect to it, just to have options like this available.