The pivot foot is the foot a player keeps on the floor while he holds the ball and turns on it. The other foot can step in any direction as often as he likes. A front pivot turns him chest first; a reverse pivot turns him back first.
The rule
FIBA’s travelling rule is Article 25 of the Official Basketball Rules. As FIBA quotes it in a refereeing explainer, a player who catches the ball while standing with both feet on the court may use either foot as the pivot, and the moment one foot is lifted, the other becomes the pivot foot. To pass or shoot he may jump off the pivot foot, but neither foot may come back to the floor before the ball is released. In that explainer, a player who lifted his pivot foot and put it back down before his shot had his basket taken away.
Other rulebooks word it differently, especially for a player who catches the ball while moving, so teach the rule your league uses.
Why coaches teach it early
A player who does not trust his pivot foot picks up the ball and freezes. A player who does can pivot away from pressure, pivot to see the floor, and pivot into a shot. The pivot is also the start of most post moves, from the drop step to the up-and-under, and of the box out, where the defender pivots into his man.
Drills
The jump stop, pivot, pass drill alternates front and reverse pivots on every rep. In two-on-one keep-away, the passers may pivot but not dribble, so the pivot becomes the way to make a passing lane. The four-corner pivot drill adds timing with four lines.