When do I win a combat? Do I have to kill a unit, or can I also move an enemy unit away during the showdown?

Yes, you can win combat by moving enemy units away during the Showdown — you don't have to kill them.

Here's how it works: you win a combat when you are the only player with units remaining at that battlefield during the Determine Combat Result step (Rule 466.3.a). That condition is checked after the Showdown closes, combat damage is dealt, and the Combat Cleanup (heal + recall) finishes. If your opponent's units are gone by then — whether killed, bounced to hand, or moved elsewhere — you win.

But you can't use Standard Move to do it. Standard Moves are prohibited during a Showdown or Combat (Rule 144.1.c). You'd need an action card or activated ability that moves an enemy unit (e.g., "Return target unit to its owner's base") and is legally playable during a Showdown (Rule 347.1). These effects are allowed during Showdowns as long as the card/ability is properly timed.

One catch for Attackers: during Combat Cleanup, if Defenders are still present at the battlefield, your Attackers automatically get Recalled (Rule 466.1.a.2). If that happens, the result becomes "No Result" — neither player wins (Rule 466.3.d). So as the Attacker, to actually win you need to remove all opposing units before the Combat Cleanup recall step. As the Defender, if you remove all Attackers (kill or move them), you win cleanly since the Attacker recall only applies to Attackers, not Defenders.