Half guard or half mount is a ground grappling position where one fighter is lying on the other, with the opponent in the bottom having one leg entangled. This position is refered to as "half guard".
The half guard is the position that is in between a full guard and side control or full mount. The combatant on top will try to unentangle the leg and obtain side control or mount, while the bottom combatant will try to transition into a full guard or alternatively attempt a sweep or submission.
In self defence the fighter on top is said to be in a better position due to the fact that he can attemp elbows and punches as well as submissions. Although most atheltes tend to pass from half guard to side control in order to strike alot more effectively without getting swept or countered.
Below is a photo of the half guard position
Effective and well known sweeps from the bottom are Old School and New york, (video of them up in the next few weeks.)