MRSA stands for Methicillin-resistant Staphylococcus aureus. MRSA is a form of a staph infection. The two initial words of that title are the important ones.
Methicillin-resistant means that this is a form of staph infection that is not cured by methicillin-based antibiotics. It means that penicillin, amoxicillin and similar drugs do not kill the MRSA bug.
MRSA seems like a recent scare, but it is not. Although the media has only taken an interest in the last few years, MRSA has been around for decades.