Specifically, it is a US Army "general service" button for Enlisted-men's ranks (private corporal, sgt.). That design was in service from 1875 to 1902, when it was replaced by the "Great Seal" button, which is still in service today.
You asked about how to clean your brass 2-piece button. There are many methods and just as many preferences. Some people clean them only with a toothpick. Others use lemon juice, or warm peroxide, or a solution of ammonia diluted 50-50 with water. If you use a liguid other than pure water, try not to submerge the button in it, because the liquid will get inside the button and thus may corrode it from the inside out. (I pour just enough liquid into a Gatorade-jug's plastic lid to make a shallow lake, and carefully place it face-down so the liquid doesn't get into the seam around the back.