Acoustic Guitar-
The acoustic steel string guitar has been used in many styles of popular music including folk, pop, rock, blues, ragtime, country and western, bluegrass, etc. This type of guitar is played frequently with both plectrum (flat pick) and fingerstyle ("finger picking") techniques. That bright jangly strumming sound you hear so often is most likely a steel string acoustic.
Classical Guitar-
The classical guitar is a nylon string guitar. Classical guitars tend to have a beautiful sound which is more mellow than their steel-string cousins. The wider string spacing helps to accomodate the intricate finger patterns performed by the plucking hand. Some people think of them as "Spanish" guitars. They are very close relatives of flamenco guitars. They generally have wider, flatter fretboards and slightly smaller, more rounded bodies than the average steel-string guitar.