A regular cassette deck has 2 heads - the heads being the things that actually affect the tape.
The main one, the most well known one, is the standard read/write head. This is used for playing back what's been recorded, and for recording new stuff on the tape too.
The second head is the erase head. This is located just before the read/write head (with respect to the tape's movement) and wipes the tape in preparation for recording new material. If the erase head wasn't there, and you tried to record new stuff, you'd end up with the new stuff, but still being able to hear the old stuff in the background. The erase head is moved back away from the tape during playback.
So what's the third head?
The third head is a read-only head, located shortly after the main read-write head (see smartalix's addition below) (with respect to the tape's movement). You find it on semi-professional and audiophile tape decks. (I only say "Semi-professional" because conventional tapes are rarely used by professionals nowdays.)
When recording something, you can hear what you're recording. But can you really know what's actually been recorded on a tape? With a 3 head deck, you can. The deck can be told to record its input on the main read/write head, but then playback from the 3rd head and send this to its output. You can then put this through your amplifier and listen to it. But you're not listening to the incoming signal, you're actually listening to the playback from the tape. So you know that everything's working - levels are right, the tape is intact etc.
smartalix pointed out to me that the main head in a 3 head deck is write-only, and never reads, and all playback (whether conventional or during recording) is via the 3rd head. This means that the head gap can now be optimized for playback on one (narrow gap) and record (slightly wider gap). This results in an improved frequency response.