Warframe

This is very easy, All you have to do is run over to a teammate when he starts to die ( and it tells you so) then hold down the button it tells you to and he will revive, after doing this 10 times, you get the achievement.

Healer

This achievement is done by reviving 10 allies, does not have to be unique revives, you can revive 1 person 10 times and obtain this achievement. To revive an ally while they are downed, run over their body and hold the B button, be careful if they are surrounded by enemies, unless you have an ability to clear a group or blind ability you will most likely die.

Something worth noting is that if you assist in a revive (multiple people can revive 1 person) that does go toward this achievement, so even if someone is reviving a person, join in quickly to get another 1 towards the 10.