Carol,
A veteran unit was usually made up largely of men who had already served one enlistment and had reenlisted. If enough soldiers reenlisted, and I'm not sure what the required percentage was, a unit could become a "Veterinized" unit. An example of one Civil War unit that "veterinized" was Company K, 3rd Kentucky Volunteer Cavalry U.S Army. Many of the soldiers in this unit reenlisted and it became Company K, 3rd Kentucky Veteran Volunteer Cavalry, U.S.Army. I've also noticed that in many instances, draftees and substitutes were integrated into these organizations, especially late in the war.
Bob