How can you tell if you have a Peace or Morgan silver dollar?

Date and design.

Morgan dollars were struck from 1878 to 1904 and again in 1921. They have a picture of Miss Liberty on the front and an eagle with spread wings on the back. The design is sort of "busy" and has a very old appearance to it.

Peace dollars were struck from late 1921 to 1935. They have a much more modern look, with the Word PEACE on the back and an eagle with folded wings.