How do you use Sketch?

Question

I've never been absolutely clear on how it works or what it does. Can someone explain?

Answer 1

Credit goes to Smogon.

Wild Smeargle Method

The Wild Smeargle Method involves leading with a Pokemon that knows the move you want and finding a wild Smeargle.
Using a Pokemon with the move you want as the lead, attack the wild Smeargle with the move. Note that you must be faster. The wild Smeargle has no choice but to Sketch the move.
Switch to your own Smeargle, remembering that you must be able to take whatever attack the wild Smeargle now knows. Use your own Sketch to copy the move on the following turn.

Answer 2

It's pretty simple. Just bring Smeargle into battle and use Sketch after your target has used one move (if you move first and use Sketch it will do nothing). Smeargle will permanently learn the last move your target used. That move will replace Sketch. For example:

1. You are leading with a Flabebe and you are up against a wild Roselia.
2. You switch to Smeargle. Roselia uses Poison Power.
3. You use Sketch, and you move before Roselia. Since the last move Roselia used was Poison Power, Smeargle will sketch Poison Power and will learn it permanently.

Notice how I said target up above; in a double or triple battle, you can have Smeargle target a teammate and learn the last move they used. This can allow you to choose what moves Smeargle learns with Sketch.