A bobber is a "bob-job". To bob something in the earlier part of the 20th century meant to cut it. When women in the '20s started cutting their hair short, the style was called a bob. In the '30s, '40s, and '50s the first and easiest way to get more power was to cut the weight. In todays dollars, for intance, many thousands of dollars worth of Indian skirted fenders went to the dump. So, when Harleys and Indians (mostly) got a bob-job, they became bobbers- the precurser to the "chopper". The chopper philosophy was basically the same, but went in a huge circle. You started by getting rid of (chopping) as much crap as possible. People started a weird combination of getting into technology and taking drugs and the bikes of the '60s and '70s started becoming more of a high-powered fashion statement. Some of the old choppers forks were so long and the steering angle so far out there that some could just barely make it around a 90 degree corner in town. This has turned into the Hog/ "factory custom" (can you spell oxymoron?) where it seems the more crap you can screw on the better.
You've probably seen pictures of old bikes that had the insignia of WWII bombers painted on the tank with very abbreiviated or missing fenders. Some custom Jap bikes could be called bobbers for the same reason. After a car hit me, I turned my bike into a bobber. I love it.