A track is a portion of the magnetic tape that is recorded onto. A full-track tape recorder uses the entire width of the tape to record a single track. Early tape decks would use full track heads although for modern day purposes, a full track machine has limited use (and is hard to find).
The more width of the tape that you use per track, the better quality recording you can get. All recording tape, especially low-end brands or tape that has been played repetitively can suffer from uneven oxide on the tape surface. This results in uneven sound levels or dropouts where the sound level drops for a fraction of a second due to a thin oxide layer. The wider the track on the tape, the less likely you are to hear tape flaws as a dropout may only exist on a portion of the width of the tape. Similarly, a tape recorded at a fast speed is less likely to produce audible tape flaws.
Half-Track: A half-track deck has two tracks, each using half of the tape width. Typically a half-track deck would be used for pro or semi pro use, and the tape can only be used in one direction.
Quarter-Track: This is the most popular format for home stereo use. The tape heads are configured so that one-quarter of the tape is used for the left and right channel, and set so that you can flip the tape over, and record two more tracks using the remaining two quarter tracks.
Four-Track: This can also be called four-channel recorder. It uses a quarter of the tape width per one of four tracks, and has the ability to record all four tracks in one direction. Quadraphonic tapes from the 1970s use the full width of the tape, as do four-channel tape recorders, typically used in home recording studios.
Multi-track: These decks can have up to 24 tracks, and tape as wide as two inches. Most of these machines are used in recording studios, although the boom in home recording studios had a number of companies produce decks that provided as many as eight individual tracks on a quarter-inch tape, and some semi-pro recorders with up to eight tracks on half-inch tape.
Crosstalk
This occurs when sound recorded on one channel bleeds over to another channel when the tape head picks up low-level sounds from an adjacent channel. The tape head picks up parts of the recording from the reverse side of the tape, often the bass-heavy portion of a recording. If the volume is turned up and can hear the muted bass notes or bass drum on what is supposed to be the blank side of the tape you have crosstalk. This is simply the nature of tape, in particular quart-track tape. Recording studios use the single-direction half-track tape to eliminate crosstalk.