The Gluteus Medius is primarily known as an abductor, however due to its fan shape, it has fibres that flex and extend as well as internally and externally rotate the hip. This versatility makes it an important contralateral stabilizer - controlling the opposite hip and pelvis in a single-leg stance during mid-gait while running and walking. It helps to prevent the dip (or hike) of the pelvis which is why you feel it more in the standing leg when doing single-leg work. An under active or weak gluteus medius can cause problems elsewhere, leading to an over-worked lower back, an irritated IT band and improper stress on the knee and hip. This is why this muscle is so heavily emphasized in rehab programs.
Here is a beginner series of open-chain exercises to help target this muscle.