A veteran is a person who was experienced in a particular background, most often referring to one with prior experience in the military and or combat.
As of 2285, Nyota Uhura was a twenty year space veteran, leaving her Old City Station co-worker confused as to why she chose such an undesirable duty station. She explained that she liked the peace and quiet of her assignment. (Star Trek III: The Search for Spock)
In 2366, Jean-Luc Picard expressed to William T. Riker that if the government of Angosia III survived the night, he would offer them Federation assistance in re-programming its veterans. (TNG: "The Hunted")
In 2368, Geordi La Forge praised Kila Marr's ability to input information into USS Enterprise-D's computer, saying she handled the unit like a veteran. (TNG: "Silicon Avatar")
Kira Nerys and members of the Shakaar resistance cell operated during the Cardassian Occupation of Bajor and participated in many combat actions. (DS9: "Shakaar", "The Darkness and the Light", "Ties of Blood and Water")
Li Nalas was a revered war hero of the Bajoran resistance during the Cardassian Occupation of Bajor. (DS9: "The Homecoming")
Miles O'Brien as of 2372 had served in 235 combat encounters and had been decorated by Starfleet fifteen times and was considered an expert on starship combat. (DS9: "Rules of Engagement")