My mom bought an iPhone in amerika and brought it to austria. Now I want to know if I just have to unlock it?
It depends on whether she purchased an unlocked iPhone, sold only by Apple themselves directly, or purchase a locked iPhone. The receipt should indicate if the iPhone is locked to a carrier. If it is locked, then as Dave said, only the carrier to whom the iPhone is locked can unlock it, and only AT&T will do so.
are there any problems known?
The biggest problem is that you will get no support nor service from Apple in Austria for a US-sold iPhone. Should a problem ever arise, you will have to ship the iPhone to a friend or relative in the US to take or send to Apple for service.