I was trying to change the bulbs on my ceiling fan fixture. When I removed the bowl to change out the bulbs the black and white wires that were attached either disconnected or tore apart. Is there anyway I can fix this problem myself? What do I need to do?

The lamp socket has loosened and has been turned over the years, and the wires have now been damaged.

If you can repair it is up to you and your abilities.

To repair this may be as simple as removing the light kit and replacing the socket with new wire leads, or one with wire terminals and re-connecting the existing wires.

Depends on if the existing wires are long enough.

Of course, the obvious caution, before you do any repairs, be sure the circuit breaker is turned or fuse is or removed before doing any work on this.

Also,
Many light kits black and white connect to the Fans blue and white.
If no blue is present, look for a black wire with white tracer, many fans use a solid black wire for fan.