How do you Stutter-Attack?

The act of cancelling attack animations to kite or chase enemies is called Orb walking. I think it first came up in the early stages of DotA and is actually a misnomer, since Orb walking had an entirely different meaning in DotA. However, the name nevertheless carried over into League of Legends.

If you're looking for a few hints to learn how to orb walk, have a look at this article from Team Dignitas.

1st Possibility:

You press "a" (standart key for attacks) and you will see a circle with your auto-attack range. You're now able to left-click somewhere in the circle, and your champion will immediately attack the closest target to the point you clicked at. Let's assume this is a champion that wants to chase you. You will press "a" and left click in his direction to attack him, and after your auto-attack, you will want to immediately right-click away from him, until your auto-attack is "up/available" again. Repeat these steps to, as mentioned above, avoid his/her auto-attacks, keeping your damage output high and reposition yourself to maximize your efficiency in a fight.

This is especially easy if you have an item that slows the opponent's movementspeed, like red buff or Frozen Mallet (really nice item on an ADC btw :3).

2nd Possiblity (not the one I prefer, but depends on summoner's choice!):

You can hold down "SHIFT" while right-clicking, your champion will attack the closest target to the point you have clicked at again, but will not show your range circle. Immediately release shift after your auto-attack hit the opponent and normally right-click away from him. Repeat, same as described above.

3rd Possibility (totally not viable for fights, but the way MOST PEOPLE I KNOW DO IT):

Right-click a target to attack, and right-click to walk away. Let me tell you why this is incredibly inaccurate and more difficult than it should be. You have to right-click your target exactly on point, which can result in missing it and even RUNNING TOWARDS IT, which will result in disadvantage and probably even death without fulfilling your role as damage dealer.

Your assessment is correct. What you're referring to is usually called orb-walking or stutter-stepping, and it's accomplished by taking advantage of the "cooldown" time between basic attacks.

Essentially, your attack speed determines how often you can send out basic attacks. But obviously you don't need to be standing still for the entire duration of the attack--you can move in between when your attack hits (or when the projectile is fired, in the case of ranged champs) and when your next attack would be ready.

As far as how people do it so fast, the short answer is practice. I've been playing League for over 2 years and I still don't have a good grasp on stutter-stepping at high attack speed. It's tough, and it requires very quick, precise clicks. If you want to learn quickly, the best strategy is probably to go into a custom game and just practice it on minion waves to start out. Then after a while, buy some attack speed, and try it again.

It first requires some time with the champion as their attack animations are all slightly different. When you get used to it you can cancel the animation by moving somewhere else. It's easier with higher attack speeds but once you reach 2 or more attack speed it becomes much harder because you would have to click faster. Hence kiting with Kog'Maws barrage at 5 attack speed is impossible. This is also possible with Melee champions.

Something that helps very well is the attack move command. When you use that command you will always attack an enemy if there is one in range. It prioritizes the target that is the closest to your cursor. If there is only a single target it doesn't matter at all where you click, as long as said target is in range. By alternating between the move and attack move commands you barely have to move your mouse while attacking.

Attack move is bound to shift + right click on default, but I moved it to my A key as it's only a single key press and my hand is there anyway. I started using it to try it and I am now addicted to it. I even use it on Melee champs to make sure my clicking frenzy hits.

Go to settings and set the action "AttackMoveClick" to "A" key, so you'll have to alternate right-click for moving and pressing "A" for attacking ... don't forget to disable auto-attacks, so in that way you won't loose any farm.