What is a bobber?
A bobber is a custom motorcycle that has had all nonessential parts, particularly the front and rear fender, removed or shortened (i.e. “bobbed”). Some call these straight bobbers to distinguish them from custom bobbers.
A custom bobber is basically a motorcycle that looks like a bobber, but has had parts added or changed instead of just removed. If the motorcycle’s frame is altered, it’s called a chopper. Other names for bobber-looking bikes can be short choppers, short chops or bobber style motorcycles.

Harley-Davidson custom bobber by Motorcycle Rehab
For a more detailed explanation, go to Bobber Republic or simply read the Wikipedia article on bobbers.