The star is the standard medallion, or "drop," of a Grand Army of the Republic (GAR) membership badge. It would have been presented to and worn by only a member of the GAR, an organization for Union veterans of the Civil War. At its height, in the 1890's, the GAR had over 400,000 members. There were also other, smaller organizations for Union veterans which competed with the GAR for members. Confederate veterans had their own organization, the United Confederate Veterans, or UCV.
Several versions of the badge, which was patterned on the Congressional Medal of Honor, were issued. All are very similar at first glance, differing only in details such as the exact angles of the points of the star, the types and numbers of symbols on the reverse, etc. The same drop was used on officer's badges. It was generally not used for the annual encampment (reunion or convention) badges, although some do have similar stars.
Anyone interested in learning more about the GAR and their badges should check out the links which I posted above. There are also several good references on the subject, the most recent and readily available being Membership and National Encampment Badges of the Grand Army of the Republic, 1866-1949 by Kenneth R. Johnson and Jeffrey B. Floyd.
Here is an example of a complete membership badge:
