Actually, there are a number of 3D-printing technologies, some of which can print across spaces and some of which can not. Each technology has certain advantages and drawbacks.
The most common technology you see nowadays is FDM, which is basically an automated glue gun. This technology is usually not capable of crossing gaps (small gaps and some very specific situations excepted). It is capable of some overhang, but you have to be careful with that, otherwise you will see sagging or even collapses. In professional printers sometimes two materials are used, one of which is dissolvable. The dissolvable material is used for printing supports, on which the non dissolvable material can be printed. Afterwards the dissolvable material is removed by dissolving it, leaving only the desired bits. Note that entry level printers can rarely do this and that objects are typically designed to work around this limitation. Advantages are that the technology is cheap in both machine and material costs and that it is capable of printing hollow objects. Printed products also have good mechanical qualities. A disadvantage is that the resultion is typically low, or it becomes a really slow process (these are inversely related to each other).
Another popular technology is SLS, which is a laser sintering powder together. It does this for one layer, then the machine deposits another layer across the print bed and then sinters parts of the new layer onto the previous one. You understand that this means that parts are supported by the powder while printing. This means it is not restricted by gravity. At the end of the print process, the powder is cleared out and usually recycled. Advantages of this technology are that it is moderately cheap, has a good resolution and produces nice, strong and clean products. Disadvantages are that models can not be hollow without an opening to the outside world (the powder can not get out) and that the plastic can be a bit brittle (not food safe and absorbs substances).
A third technology is SLA, which uses projected or laser light and a light sensitive resin. By directing a couple of intersecting beams to a certain spot, the plastic on this spot is cured and hardened, leaving the rest a fluid. For overhangs usually temporary supports are used. Advantages are really high resolutions, making for really smooth and detailed objects. Disadvantages are that the price of the curing resin is typically high and that it has limited mechanical qualities. Hollow objects are also only possible with openings to the outside world, as the fluid needs to be removed (and is expensive).
There are more technologies, but these are the most common ones. Please read about them on the Wikipedia page (
http://en.wikipedia.org/wiki/3D_printing#Additive_processes). 3D-printing is one of my pet subjects and I have worked with or seen quite a number of these technologies (LOM, PP, SLA, SLS, FDM, DMLS and silver through FDM and lost wax) so if you have any additional questions, ask away!