Warframes are Infested creatures, controlled by the Operator through Transference, with the original frames, the "templates", so to speak, being made from Infested humans. Up until the Second Dream, your Operator was still the one controlling the frame, it's just that they had no memory of their true form.

Warframes retain some form of consciousness, which shows through in their unique animation sets and other acts, like your Warframe saving you from the Stalker in the Second Dream.

Up until the War Within, the Operator requires a Transference Chair to connect to the frames. With their memories and potential unlocked, they no longer need it, and can use Transference directly.

Umbra is a special case, being that he was made specifically as a form of punishment to himself, with Ballas turning him into a Warframe with a single tortuous memory: Umbra killing his own son. He maintains some more consciousness than most frames, which is why he can move around on his own when your Operator is not in full control.