Pivoting is the ability to go from one skating direction to another without losing momentum. In order to do this, you'll perform what is known in figure skating as a "mohawk."
I like the word "pivot" because a mohawk in figure skating is a very precise maneuver, but for ice hockey, the mohawk is applied in more "open" manner (i.e. your legs and feet don't have to be close before you execute this move). A pivot is simply turning your upper body and shoulders from the forwards direction to the backwards direction. If you turn left, then you will pick up your left leg and skate, then plant it back on the ice, facing in the other direction. Your feet will be pointed in opposite directions for a brief second, but what should happen next is that as you continue turning your upper body to the left, you will pick up your right leg, and point it in the same direction as the left leg. That's a pivot!
Clearly there are three variations (forwards, going left then right, then backwards, going left then right).
I once watched a Bruins game, and the Bruins defenseman Zdeno Chara was skating backwards (fast) guarding an onrushing forward. The forward turned slightly, and skated between Chara and the boards. Instead of turning towards the forward, Chara turned in the opposite direction. The forward gained an extra second to get past Chara, and (I think) he scored. I remember the pivot because the commentator said "If Chara could have pivoted towards his man, he could have closed down his angle faster." I have no doubt that Chara could pivot in that direction (backwards, turning left), but it was a moment in an NHL game in which this "advanced" move was revealed.