An internship was a period of time during which someone studied and practiced their chosen career while being supervised and mentored by a more senior person in that field.
Sometime before 2268, Dr. M'Benga interned on a Vulcan ward. (TOS: "A Private Little War")
In 2352, Beverly Crusher did a medical internship on planet Delos IV under the supervision of Doctor Dalen Quaice. (TNG: "Remember Me")
In 2369, Amanda Rogers was selected out of hundreds of applicants for an internship on the USS Enterprise-D (TNG: "True Q")
In 2375, Julian Bashir arranged a internship position for Sarina Douglas at the Corgal Research Center. (DS9: "Chrysalis")
The episode's script gave the location of the center as Corgal II.