The exposition of a classical piece in sonata form is analogous to the hook of a movie: it grabs the listener's attention, introduces the main ideas (themes) of the piece, and gets the listener ready for more.
It makes up the first two A's of the AABA interpretation of sonata form. Inside that A, which is repeated twice, there is generally an internal structure of ABA. Many times the second of the two A's there is played in the dominant rather than the tonic.
Of this internal structure, the A theme (again considering the common case) has higher energy, while B is more calm and relaxing (which can be, and is, interpreted in many ways), but that doesn't necessarily imply lower tension. In fact, it usually has higher tension.
Sometimes this internal ABA is padded with what some would call C's and D's, being internal developments. These are technically not themes; rather, tools to manipulate the tension between themes.
Consider allegro con brio from Beethoven's Fifth Symphony (the popular one): The initial four-note theme and its variations for the first thirty seconds (or so) make up the A part, then the french horn leads into the calmer B part. After that, A is repeated again in a major key, and a cadence leads back to repeat it all again. This makes up the exposition.