So after you beat Batman Arkham City on either normal or hard settings, you will be able to play through the game in the "New Game Plus" mode. This mode allows you to play the game again without losing your gadgets or the XP that you earned through your first play through.
Only catch is that all the bad guys are going to be a lot tougher the next time around. They won't have the typical counter symbol over their heads like it usually does. They will also be tougher and in different configurations. Also the bosses will be much harder to defeat.
Don't worry thought. If you aren't interested in ripping out all of your hair from the stress of getting your ass kicked in this "New Game Plus" mode. It's optional. But replaying without it won't keep your gadgets and XP.
source:Arkham City Community Page