You can easily find Silver, Copper, Brass and Stainless Steel cleaners in stores across the country.
You should clean with soft cloths, but if it is a real dirty outdoor mailbox, you may want to use a nylon scrubbing cloth to help out. Just be careful, it makes more scratches than you think, and they do show up in the shinny brass.
Try cheese cloth as the final polishing cloth after cleaning.
For some decorative metal pieces, like brass wall plates, after you clean it up spray it with clear laquer. That prevents the oxidation that starts the discoloration. When you need to get the laquer off (because the little cracks you missed are beginning to turn black or green, just use a laquer stripper to start all over.
The Cobra brass cleaner people asked me to test their product against their claim that you don't get that black residue while cleaning. It is true, and it is a good cleaner.