Half guard is a transitional position where you trap one of your opponent's legs between both of yours. The top player has passed one leg through your guard but not both. It sits between full guard and side control on the control spectrum.
From the bottom, half guard allows sweeps, back takes, and guard recovery. From the top, the goal is usually to free the trapped leg and complete the pass to side control.
Half guard is common in live rolling because closed guard often breaks into it when the passer manages to step one leg back. Learning to make half guard active rather than passive is a key development milestone for beginners.
For attacking and defensive concepts, see Half Guard.