I was born in Germany in 1991 on a German mother and a British father. I've lived in the UK since I was a couple months old and I am now 19, because we lived in the UK my mother didn't want things to get complicated with aplying for passports so she only let me have a British one instead of a German one. I heard Germany doesn't recognise dual nationality, is this still true? I didn't think it was.
Do I qualify for a German passport? If I do, would I be able to keep my British one or not?

Yes, if you have one German parent, you have German citizenship, regardless of where you were born.

Source: http://en.wikipedia.org/wiki/German_nationality_law

A child born to an American parent and a German parent acquires both American and German citizenship at birth, regardless of place of birth. Neither country requires a person born under these circumstances to choose between American and German citizenship. They may keep both for life.