Definition: A testing mission is the purpose a piece of testing serves: the information it's meant to provide, and for whom. Different missions call for different approaches, such as finding as many important bugs as possible quickly, or showing stakeholders what a product can do.
So what? Being clear on the mission tells a team which testing approach fits. A bug hunt suits a mission to find lots of bugs fast, but not one that needs deep, careful coverage of a risky area.
Examples: Giving product managers information to help them make a release decision. Running a session to demonstrate a product's capability to key stakeholders before go-live.