In military and other covert operations, terminate with extreme prejudice is a euphemism for aggressive execution (playing on the expression "termination with prejudice" of an employment contract).
In a military intelligence context, it is generally understood as an order to assassinate. Its meaning was explained in a New York Times report on the Green Beret Affair (an incident during the Vietnam War), and subsequently popularized by its usage in the film Apocalypse Now.
The expression may also be used as a straightforward statement of intent in some jurisdictions to refer to more overt summary executions, i.e. in the context of a 'death warrant' or 'kill order' issued as part of a 'shoot on sight' or similar policy.