You have to complete an entire episode (all the chapters within) without dying on extreme mode. You only unlock extreme mode after completing the game on any difficulty. The prologue mission counts as an entire episode and only lasts around 8 minutes. Choose the healer units when the option arises. Protect the large ship as it makes it's way to a set location. When you get your special, feel free to use it () when you're low on health or the ship you're protecting seems to be taking a lot of damage. If you die for some reason, you will need to restart the entire mission over.