half h bridge is what the industry refers to a push-pull driver. this kind of thing can both source and sink. easiest to think of it is a npn and a pnp, with bases in parallel and emitters shorted.
a full bridge has two push-pull drivers. turning one to sink and one to source yields a motor direction, reversing their logic reverses motors directions. used to reverse current direction.
a dual h bridge has two h bridges, or four push-pull stages.