adjective that's a bit difficult to use properly and is (or was, when I was in school) mostly used by teenagers while adults consider it improper for polite company
The primary meaning (when used on people) is sexually attractive or horny. However, the meaning has extended to become an all-purpose positive along the lines of cool, fab and phat when used to describe things, events, skills, etc.
One of my schoolteachers (who was pretty cool) once told the class that in talking to us, the word had leaked into her daily usage, earning her strange glances from friends and family.
The short rundown: better don't use it with anyone older than 30 or wearing a suit, but most younger people won't mind and should you have a German girlfriend or boyfriend, they'll probably like being called "geil".