A V-engine has its cylinders arranged in two banks, with a common crankshaft. As a result the engine is a 'V' shape - hence the name.
V-engines usually have small overall size for their displacement, which makes them easy to package within a vehicle.
V-twin engines usually have their two cylinders at an angle of 90 degrees. V4s with a 90 degree angle have been built, but only provide reasonable balance (and therefore smoothness) when provided with a balance shaft.
V6 engines commonly have a 60 degree bank angle, though 90 degree V6s have also been built - often for packaging reasons, as the 90 degree angle reduces the height of the engine.
V8 engines almost always have a 90 degree bank angle, and V12s are usually 60 degree - though as with V6s, the bank angle is sometimes altered to improve packaging.