To abandon something is to give it up completely. If you’re in a cabin and a forest fire approaches, you’d better get in your car and abandon your cabin, or else be prepared to abandon your life.
If you abandon something, you let it go, so when you describe someone as "acting with abandon," it means they have let go of restrictions or inhibitions. Maybe you like to dance with abandon in the privacy of your own room. But abandon could also mean people have abandoned their senses. To "drive with abandon," for example, means to drive recklessly.