Generally, you will hole out with the same ball you play from the teeing area. You can always use a new ball when starting a hole. You can also substitute a different ball any time you are taking relief, including both free and penalty relief. Unless the one-ball Local Rule is in effect, the substituted ball could be any brand. On the putting green however, when you mark and lift your ball, you must replace that same ball to finish out the hole.
When your ball is lifted, you can almost always clean it (except for a few specific circumstances). If you happen to run out of balls, you can borrow one from any other player, including a practice or X-Out ball (which are generally conforming balls).
If you play a wrong ball, you lose the hole in match play or get a two-stroke penalty in stroke play. In stroke play, you must correct your mistake by playing the right ball or otherwise playing under the Rules (for example, by playing under stroke and distance if your ball is lost).