(covers information from several alternate timelines)
Academic suspension is a sanctioning of students at Starfleet Academy for violations of the institution's regulations. It can sometimes be imposed temporarily on a student while an inquiry is undertaken to determine if they broke such regulations, but is usually applied when a student is found to have done so. It involves a student being unable to sign up for classes, and unable to be assigned to training crews on starships.
In 2258 of the alternate reality, James T. Kirk was placed on academic suspension during the inquiry into his modification of the Kobayashi Maru scenario. The charges were brought against him by Spock, who at the time was an instructor at the Starfleet Academy and in charge of programming the scenario. Because of this, he was not assigned to a ship during the distress call to Vulcan. Instead, Leonard McCoy infected him with a vaccine to make him appear ill, and took him aboard the USS Enterprise as his patient. (Star Trek)
In 2368, Wesley Crusher, Jean Hajar and Sito Jaxa were suspended from Starfleet Academy following a training accident around Saturn as part of Nova Squadron. They attempted an illegal maneuver, the Kolvoord Starburst, and the resulting accident cost fellow squadron member Joshua Albert his life. It was found that the leader of the squadron, Nicholas Locarno, had convinced Crusher, Hajar and Sito to cover the incident up. Locarno was expelled for leading the coverup. (TNG: "The First Duty")