The kite

Attack, step, attack: hit without getting caught.

From GoldBot

Professor Zéphyrinthe Academy's instructor

« Kiting is a waltz: the attack leaves, then you step. Move too early and you cancel it; plant your feet and they catch you. »

Hit, then step
You attack, and as soon as the attack leaves, you take a step back. Then again: hit, step, hit.
Always at max range
The gap doesn't change: he stays at the edge of your range, never in his. He takes every hit and lands none.
Forwards to chase
He gives up and runs: same rhythm, forwards. You stay in range and finish him.
The classic mistake
You stand still to attack: he walks up and reaches you. Or you run without attacking: you deal nothing. Either way, you lose the duel.

When to do it

A melee champion runs at you: you back off between your attacks, and he never gets there.
An enemy runs away: same rhythm, forwards: you stay in range and keep hitting.
In lane, against a shorter range: a step in to hit, a step back out of theirs.

Attack, then move: as soon as the attack leaves, click to move; attack again as soon as it's ready.
Use attack-move: one key attacks the enemy closest to your cursor, no misclick on the ground.
Don't cancel your attack: moving too early cancels it. Let it leave, then move.

The classic mistakeStanding still to attack: he reaches you. Or running without attacking: you deal nothing. The kite is both at once.

Good to knowAgainst a diver faster than you, clicks aren't enough: keep a slow, a dash or your Flash to get away.

Iron to Silver

Start simple: one attack, one step back. Practice it in a custom game until you don't have to think about it.

Gold to Emerald

Bind attack-move and use it for every attack: fewer missed clicks, and you hit the enemy closest to your cursor.

Diamond and above

Kite at the edge of your range, not further: every step too many is an attack you don't make. The best kite loses no damage.