Touristenfahrer at the Nürburgring
Anybody can drive at the Nürburgring in their own car. The circuit is usually open from about March until December, and is available to the public almost every day that it isn't in use for some other purpose. During the week the Nordschleife is usually hired out to the motor industry or private motor clubs for their own use, but even on these days the track is often open to public for a couple of hours in the evening.
For a full list of opening times go to the official Nürburgring website.
If you've never been before but are planning a trip then I would advise reading Ben Lovejoy's comprehensive Nürburgring website, and also the General Terms and Conditions for Driving on the Nürburg Ring, below.