A staph infection is a short form for the organism that is called Staphylococcus aureus (or sometimes epidermis) which is a very common "gram positive" organism. Normally, it is around and doesn't affect you, but if it gets in to a cut and infects it, it can actually be pretty nasty. Staph can be both really easy to treat, but can be difficult at times. You may have heard the term "methicillin resistant staph aureus" or MRSA...this is the nightmare of doctors as this critter is resistant to pretty much all the medications that we have currently. Fortunately, it's not that common.
Usually, a drug like naficillin, cefazolin or clindamycin will take away infections with staph. I think it's a great idea that you're going to an ID specialist if the problem is persisting.