A battalion or company that is assigned to perform a specific mission is called what?

The term that accurately describes a battalion or company assigned to perform a specific mission is "Task Force." A Task Force is a military unit formed to accomplish a particular objective which may consist of personnel from different units or branches that are brought together for a unique mission. This allows for flexibility and the effective use of various capabilities in a cohesive manner.

In a military context, a Task Force is temporary and often dissolves once the mission is complete. It is distinct from a detachment, which usually refers to a smaller unit set apart for a specific purpose but does not necessarily have the same level of command and control or mission focus as a Task Force. A unit generally refers to a basic organizational element in the military structure, while an element can refer to any part of a military organization, not specifically tied to a particular mission.