Nonsteroidal anti-inflammatory drugs, usually abbreviated to NSAIDs or NAIDs, are drugs which offer anti-inflammatory effects. Ibuprofen and aspirin are two examples of NSAIDS.
NSAIDS are used for acute and chronic pain and although ibuprofen and aspirin both sound fairly normal and safe, the widespread use of NSAIDS has made the potenitial side effects increasingly prevalent. The main adverse drug reactions associated with use of NSAIDs relate to direct and indirect irritation of the gastrointestinal tract. You most likely will be or already are prescribed an additional pill alongside your NSAID, which is there to help balance the effects of the other drug.
There is a lot of information on NSAIDS here http://en.wikipedia.org/wiki/Non-steroidal_anti-inflammatory_drug