A fellowship is a grant given by an academic institution to a scholar for research or study. Usually, receiving a fellowship required one submit an application.
In 2259, Nurse Christine Chapel informed Doctor Joseph M'Benga that she was considering applying for a fellowship in archaeological medicine that would station her on Vulcan for two months. (SNW: "The Broken Circle")
In 2371, Jake Sisko was offered a writing fellowship by the Pennington School in New Zealand, based solely on the quality of a story he had written. However, he later decided to turn down the fellowship. (DS9: "Explorers")
In the writers' first draft script of ENT: "Dear Doctor", Doctor Lucas was referred to as currently "doing a fellowship" on Denobula, at a research hospital where Dr. Phlox had done his training.