Personnel records of veterans are located in several places. For veterans who were discharged prior to 1910 most records are at the National Archives in Washington D.C.  For confederate veterans, there may be records located in the State archives of the State they were from.

For records of veterans discharged after 1910 the records are most likely at the National Personnel Records Center in St. Louis Missouri.  If the veteran was discharge in the last few years, the records may still be in the hands of the military. 

There are exceptions based on whether the person was in the Navy or was an officer etc.  As time permits I will add more on this