The vapor that comes out of the exhaust when the car is cold is normal. It's a thin vapor that usually curls up over the car and dissipates.
Sometimes it's so cold outside that the thin white vapor coming out of the tailpipes won't stop, and that's OK, too.
If there's a heavier white smoke coming out of the tailpipe, that's usually the sign of an engine problem, typically when coolant/antifreeze enters the combustion chamber because of a bad cylinder head gasket or another problem that ends up burning coolant.
There are other signs that may point to a problem, including oil and antifreeze mixing and the engine, and oil on the dipstick showing a milky consistency. Also, if you're burning coolant, your coolant level may be low. If you suspect that there may be a problem, take it into a service center. Never take off the radiator cap when the engine is warm.
Black smoke usually means that the engine is putting in more gas that it is supposed to. This could be a number of problems and quite often is hard to pin-point and repair. Sometimes it is something simple, but usually not. If this is the case, and the engine has been driven for a while in this condition, the catalytic converter may be clogged up and need to be replaced which is expensive(a few hundred dollars to replace).