I recently bought a house that was built in 1960. During the inspection, I was told that the majority of outlets weren't grounded. However, it appears a few are such as in the bathroom. How do I determine if my circuit breaker box is grounded? I had an electrician come out and he said it wasn't. Then he proceeded to tell me that he couldn't ground my fuse box and each of my outlets would need to be rewired. Somebody else told me that this doesn't sound right. I have read a lot on the internet and not sure what to do next?
It's unlikely that your service panel ("circuit breaker / fuse box") is not grounded. That could lead to all sorts of weird problems, not to mention that it would be tantamount to malpractice for an inspector and an electrician to walk away from an electric service in that condition. Furthermore you said that some outlets are grounded. If they passed inspection then you have grounded service.
(Technical clarification: usually the ground and neutral buses are connected in the main panel, and the service is grounded upstream. By code such a connection can only be made in the main panel. In this case the main panel is still "grounded" even though one might pedantically argue that the ground connection is not at the main panel.)
It sounds like you've gotten confused, so here's the simple situation: Most of the outlets inside your house are ungrounded. They're the ones with only two holes, that can't accept a three-prong plug without an adaptor. That's it, and it's very common in construction of that vintage or older.
The only way to provide grounded outlets is to run new wires from the service panel and replace the receptacles. It's usually quite a pain/expense to do that, and apparently it's not required by your local ordinances. (NB: See @wallyk's comment: You may have 3-conductor wiring to outlet boxes, in which case switching to grounded receptacles is a relatively easy project you can do yourself with the power off.)
One tip: If you do opt to pull new wires so you can ground outlets use 12-gauge wire, not 14-gauge. That way you can support 20-amp circuits, which I have found comes in handy quite often, and the marginal cost shouldn't be that significant.
In many cases in older houses the outlet box is not grounded due to code at the time. If grounded the ground was usually metal mesh around the wire screwed to the box and the neutral bar. The ground then was usually a wire that was then ran from the neutral bar to a pipe near by and clamped to it. The simple way to find out is to use a volt meter and put the red lead in the hot side of the outlet or touch it to the black wire. The other lead touch it to the metal box. If you get a reading of around 110 to 120 volts it's grounded. At which point grounding an outlet is a matter of adding a ground wire from the box to the outlet. If it is not grounded the next cheaper alternative to running new wire is to put a GFI outlet in. This basically gives the outlet its own built in breaker.