It sounds like you have tubular tires/wheels. http://sheldonbrown.com/tires.html has good information on the subject. Basically, the tires have an internal inner tube and are glued onto the rim. This used to be more common, and now is mostly seen on high end racing wheels and cyclocross bikes.

Tubulars are probably more trouble than you want to deal with; maybe you can find someone who will trade you a clincher wheelset for your tubulars?