The classical guitar has nylon strings and a wider neck than the other types of guitar. It is most commonly used for playing Classical, Flamenco and Fingerstyles. Generally it is much cheaper than other types of guitar and is recommended for beginning guitarists.
The steel string acoustic has steel strings and is most commonly played by strumming or fingerpicking groups of notes called chords. This is the type of acoustic guitar you will hear in most modern styles of music e.g. Top 40 Rock and Pop music.