A deep space assignment was a term used to describe missions into uncharted areas of space.
In 2269, Lieutenant Mira Romaine was given her first deep space assignment aboard the Enterprise, when she was placed in charge of supervising the transfer of equipment to Memory Alpha. (TOS: "The Lights of Zetar")
After graduating Starfleet Academy in 2327, Ensigns Jean-Luc Picard, Cortan Zweller and Marta Batanides went to Starbase Earhart prior to receiving their first deep space assignments. Zweller was later posted to the USS Ajax under Captain Narth. (TNG: "Tapestry")
In 2369, Captain Jean-Luc Picard, Lieutenant Commanders Data and La Forge and Counselor Troi attended a psychological conference on the effects of long-term deep space assignments. Among the attendees were Doctors Mizan and Vassbinder. (TNG: "Timescape")
In 2371, the crew of the USS Voyager ended up on the longest deep space assignment in the history of Starfleet, when the ship became lost in the Delta Quadrant as a result of the Caretaker's array. (VOY: "Caretaker", "Endgame", et al.)
Starfleet personnel assigned to any one of the deep space stations were considered to be on a deep space assignment. When Julian Bashir was first posted to Deep Space 9 in 2369, he became excited at the prospect of practicing "frontier medicine", as he called it. In 2370, Ensign Melora Pazlar was assigned to the station in what was her first deep space assignment. She conducted a survey of a sector in the Gamma Quadrant, thanks to the proximity of the Bajoran wormhole. (DS9: "Emissary", "Melora")
In 2374, after connecting to a seemingly abandoned sensor network, Seven of Nine detected the USS Prometheus, a Starfleet vessel on the outer edge of the Alpha Quadrant. When Captain Kathryn Janeway became aware of the ship's location, she observed that it was probably on a deep space mission. In fact, it had been hijacked by Romulans and was heading for Romulan space at high warp. (VOY: "Message in a Bottle")