From a private pilot's perspective, there are two possible options: a foreign-based license or a 'full' FAA pilot's license.
The first type is defined in 14 CFR 61.75, which starts like this:
§61.75 Private pilot certificate issued on the basis of a foreign
(a) General. A person who holds a foreign pilot license at the private
pilot level or higher that was issued by a contracting State to the
Convention on International Civil Aviation may apply for and be issued
a U.S. private pilot certificate with the appropriate ratings if the
foreign pilot license meets the requirements of this section.
You apply to the FAA and they check with Germany (or wherever) that your foreign license is valid. If it is, they'll issue a foreign-based license "with the appropriate ratings" with no flight or theory test of any kind required. That's for VFR only; if you're instrument rated then you also have to do a theory test as well if you want to fly IFR.
That sounds great - and I suppose it is - but there are two big practical concerns. First, any limitations on the foreign license are applied, e.g. if your license is for daytime VFR only then you're still restricted to that, even if a regular FAA license would allow night VFR. And if your foreign license expires or becomes invalid for any reason, the FAA one does too.
Second, you won't have been required to learn anything about US regulations or flying practices. So you might not fly safely or legally because you simply don't know the rules or even what services are available to you as a pilot (FSS, Flight Following, TRSA etc.). One way to handle this is to get some general orientation or introductory training from a US instructor or school, but as a foreigner that will require TSA approval unless you can do it as ground school only (which is not defined as "flight training"). But you might find - as I did - that some instructors will not work with a foreign student at all until they have TSA approval, whether or not it's just ground work.
The other option is to get a normal, full FAA license. You have to pass the same FAA medical, written and flight tests as US pilots to the same standards. You can count your previous flight time towards the required hours but you do need to be able to pass the tests so you will certainly need some instruction. And as a foreigner you will need TSA approval before you can receive any flight training.
The big advantage of getting a full license is that it never expires. If you really want to get one and you have the time and money then you could look into spending a month somewhere suitable in the US (Florida? Arizona?) at a flight school to get it as quickly and cheaply as possible.