It is what button collectors call a "Sportsman" (or, "Sporting") button... in this case, the sport of Hunting, typically showing a "large game" animal. Although there is a similar-looking State Militia button, showing the head of a stag, your button has an obviously civilian decorative border around the button's main emblem.
Neal, when you want us to help you date a button, it'd help a lot for you to tell us exactly what the button's backmark says. (Exact spelling, and any punctuation, such as stars, etc.) Often, the exact backmark is crucial for accurately time-dating a button. I think I can make out the name "Floyd" and "Extra" on your brass 2-piece button's back. George Floyd & Company of Philadelphia made gilted brass buttons from 1842-44. Info-source: the book "American Military Button Makers And Dealers; Their Backmarks & Dates" by William F. McGuinn & Bruce S. Bazelon.
