The purpose of the Infield Fly rule is to protect the runners from a fielder intentionally letting a flyball drop in order to turn a double play.
Knowing this purpose explains why we adjudicate this rule in the way we do.
The Rule Book Says
An INFIELD FLY is a fair fly ball (not including a line drive nor an attempted bunt) which can be caught by an infielder with ordinary effort, when first and second, or first, second and third bases are occupied, before two are out.
The pitcher, catcher and any outfielder who stations himself in the infield on the play shall be considered infielders for the purpose of this rule.
When it seems apparent that a batted ball will be an Infield Fly, the umpire shall immediately declare “Infield Fly” for the benefit of the runners. If the ball is near the baselines, the umpire shall declare “Infield Fly, if Fair.”
The ball is alive and runners may advance at the risk of the ball being caught, or retouch and advance after the ball is touched, the same as on any fly ball. If the hit becomes a foul ball, it is treated the same as any foul.
If a declared Infield Fly is allowed to fall untouched to the ground, and bounces foul before passing first or third base, it is a foul ball. If a declared Infield Fly falls untouched to the ground outside the baseline, and bounces fair before passing first or third base, it is an Infield Fly
Key Points
Once you know the conditions upon which an Infield Fly applies, exactly when to call it becomes the next challenge.
Taking clues from the rule definition itself and re-ordering them in the practical order they happen gives us the answer.
When it seems apparent the ball can be caught by an infielder with ordinary effort the umpire shall immediately declare "Infield Fly"
For an umpire this is when the fielder is stationary (or moving slowly) under the ball ready to take the catch.
Mechanics
All umpires can call an infield fly by pointing with their right hand in the air and shouting "Infield Fly"
If the ball is close to the foul lines, the umpires on those lines should call "Infield Fly if Fair"
Umpires use a signal with each other before a pitch and play in which an infield fly may occur. That signal is the right hand accross the chest. Clenched fist for 0 outs and 1 finger for 1 out.