Travelling: the gather step and how many steps you actually get
Rules
Where the count starts, why it looks like three steps when it is two, and the situations that catch players out.
The rule is that after you gather the ball, you may take two steps before passing, shooting or being required to have released the ball. Everything confusing about travelling comes from the word gather.
The gather is the moment you gain control of the ball with one or both hands while ending your dribble. That moment itself is not a step, even if a foot is on the floor when it happens, and even if a foot comes down immediately afterwards as part of the same motion. So a player driving at speed will often appear to take three: the foot that lands as he collects the ball, then step one, then step two. Slow it down and the first of those is the gather.
From a standing start with no dribble, the rules are stricter. You catch the ball with both feet on the floor, one foot becomes your pivot as soon as you lift the other, and the pivot foot must not slide, lift and return, or change. You may lift it to shoot or pass, provided the ball leaves your hands before that foot touches the floor again.
A few specific situations produce most of the calls. Starting a dribble: the ball must leave your hand before your pivot foot leaves the floor, and reversing that order is the most frequently missed travel in amateur play. Falling to the floor while holding the ball and then sliding or rolling over is a travel. Getting up off the floor with the ball while holding it is a travel. Catching your own missed shot is legal if the shot hit the rim or if it was a genuine attempt, but catching your own deliberate pass to nobody and continuing is not.
The step back is legal for the same reason a drive to the rim is. The player gathers, takes one step backwards, lands on the second step and shoots. Two steps, in a different direction. It looks illegal because we associate the two step allowance with moving forward.
If you want to see the rule clearly in a live game, watch the feet of a player receiving a pass while sprinting toward the basket. Count from the instant the ball is in his hands, not from the instant it left the passer's. The count almost always comes out at two, and the play that looked wrong looks ordinary.