Use fault when explaining who is responsible for something bad.

It's my fault that the car was stolen. I left the window open.

WRONG!!! --> It's my mistake the car was stolen. I left the window open. <-- WRONG!!!

Use mistake or error for talking about something that you did or thought which was wrong. Error is slightly more formal than mistake.

We lost a week's work due to a computer error.

I still make lots of mistakes in my essays.

WRONG!!! -->I still make lots of faults in my essays. <-- WRONG!!!