Translation:Her mission ends at the end of the week.
no it could not be just termine if it where it would have to be referring to an object i.e. il termine sa mission - he/it ends his/its/her mission, but sa mission se termine- his mission ends.
what is the mission ending precisely? itself, so you have to use the reflexive form.:) hope that explains it well enough:$ (also as a side-note when it's reflexive like this you have to use etre as opposed to avoir in the passé composé)