How do critical hits work?

There are two different types of critical hits: Critical melee hits, and critical arrows.

Melee crits

These occur when the player strikes a mob while moving downwards. Jumping and being stuck in a cobweb count, but moving down a ladder or sinking in water do not. Additionally, you can't do critical hits while blinded.

A critical hit has a sparkle effect, and sometimes does more damage than a regular one. The damage bonus is random, and varies between purely visual (i.e. +0) to +50% damage (rounded down), plus half a heart.

For example, a normal hit with the Diamond Sword does 7 damage (3.5 hearts), while a critical hit does between 7 and 11 (3.5 - 5.5 hearts).

Critical arrows

The critical hit effect on an arrow indicates that it was charged fully; the damage bonus for those works the same as for melee critical strikes.

you can score a critical hit by hitting a mob (passive or hostile) while falling or shooting a arrow that is "fully charged" at it. A critical hit from a Bow will deal double damage.