How does winning combat constitute? Is it conquering or killing one unit?

Neither — you win a combat by being the only player with units remaining at that battlefield when the Combat Result is determined (Rule 466.3.a).

No kills required: the opponent's units can die, get bounced, or be moved away — if only your units remain, you win the combat. No damage needs to have been dealt at all. Conversely, killing all their units is meaningless if you don't have any units left yourself — that's "No Result," not a win (Rule 466.3.d).

Conquering is a separate thing that follows winning: after the result, the player with units remaining establishes control, and that's what triggers a Conquer point if they haven't scored that battlefield yet (Rule 466.5.d). You can also Conquer with no combat at all (moving onto an empty battlefield) — that's a Non-Combat Showdown, and it is not winning a combat.