Enter the level and kill every enemy you encounter. Leave no one standing.
One of the simplest missions the Lotus can assign is to travel to an outpost or ship and kill every enemy in sight. After a sufficient number have been killed, the rest will flee and the Tenno can head for extraction.
Exterminate missions on the Orokin Void tileset feature slightly different dialogue (although the completion dialogue is the same). Exterminate missions on the Zariman tileset are handled by Cavalero.
(Grineer) "There is a large platoon of Grineer Marines stationed here. Leave no one standing."
(Corpus) "Eliminate all Corpus threats."
(Infested, variant) "I'm detecting Infested bio-signatures all over the place. You know what to do."
(Infested, variant) "The Infestation has corrupted all life here. Let's clear it out."
(Infested, variant) "We have got an infestation. Extermination is the only option."
(Infested, variant) "We need to clean this area up. Search for the source of the Infestation and exterminate it."
(Infested, variant) "The area is crawling with Infested. If it's not a Tenno, eradicate it."
(Infested) "I'm still detecting a lot of Infestation activity. Let's clean it up."
"You have broken their ranks and they are on the run. Another successful mission. Find extraction."
"The enemy is broken. Well done. Now get to extraction."
(mission start, variant) "You aren't the first ones here. I'm detecting multiple enemy signatures, and they're all under the control of the Tower. Eliminate the threat."
(mission start, variant) "The Tower's Neural Sentry is operational. I'm picking up multiple threats, all under Orokin control. Wipe them all out."
(mission start, variant) "You are not alone. This Tower is full of lifeforms, all corrupted by the Neural Sentry. Clean them out."
(during mission, variant) "Keep fighting. As long as the enemy is here, this Tower is useless to us."
(during mission, variant) "The Grineer should have known better than to walk into an Orokin Tower unprepared. Let's clean up their mess."
(during mission, variant) "The Tower's Neural Sentry has turned these soldiers into mindless attack drones. You need to take them out."