Most compressors require very little maintenance, especially smaller compressors. If it has a filter, dust it off from time to time. If it is a foam filter you may need to put a little oil on it (check your manual). Check your moisture trap and make sure it isn't full and if it has a filter in it, dust that filter off occasionally. If it has a tank you should drain that daily (it should have a drain valve on it). When air is compressed the moisture gets squeezed out and condenses in the tank. This can cause rust which will eventually weaken the tank and allow flakes of rust to blow off into the airstream (very bad for airbrushing!)
Once every blue moon I like to brush some dishwashing soap over all the connections to make sure they aren't leaking. If they are you'll see bubbles from the soap. If you have a leak, get some Teflon tape from any hardware store (in the plumbing section), unscrew the connection, wrap a single layer of Teflon tape around it, and put it back together.
Check your manual and see if there is anything else, but there probably isn't. I have two compressors and neither of them require anything other than that.