A student was an individual obtaining an education, usually at a school, college, university, or other academic institution.
The Deca were a group of top-ranked students at the Time Lord Academy. (PROSE: Divided Loyalties) The Doctor was one of them, even though he did not excel academically, graduating only with a 51% pass rate. (TV: The Ribos Operation) Romana, however, achieved a "Triple First" when she was at the Academy centuries later. (TV: The Ribos Operation)
Timothy Dean, a student at Hulton College in 1914, accidentally found the pod where the Seventh Doctor had stored his biodata while living as the human John Smith. Leakage from the pod temporarily gave Dean Time Lord abilities. (PROSE: Human Nature) The Tenth Doctor became human again in 1913 to evade the Family of Blood. The Chameleon Arch created for him the identity of history teacher John Smith at the Farringham School for Boys. One of his students, Tim Latimer, found the fob watch where he had stored his Time Lord essence. It communicated with and instructed Latimer during the Family's assault on the school. (TV: Human Nature / The Family of Blood)
The students at Coal Hill School knew about the romantic relationship between Clara Oswald and Danny Pink, even making graffiti around the school mocking it. Later, a student-teacher conference was interrupted when the Skovox Blitzer returned from the Time Vortex a day earlier than the Doctor had predicted. (TV: The Caretaker)