An assignment patch is a badge or emblem worn as part of one's uniform, to distinguish one's mission or posting. Mission patches were first used by NASA's Gemini 5 crew in 1965, and new patches were created for subsequent missions.
When uniform tunics were not worn, many crew people instead wore standard jumpsuits or other utility clothing. While such identification was sometimes omitted, simpler embroidered patterns were worn on the uniform breast. On smaller ships where the uniform of the day was less formal, such as the USS Sagittarius, a regular assignment patch could be worn on a plain jumpsuit. (TOSepisode: "The Cage", et al.; VANnovel: Harbinger)