In their closest meanings, "still" is used for a situation that is ongoing or unchanged at any point in time, or regardless of any factor, stated or otherwise:
She still loves you. If you lose all your money, she will still love
you. Even when you were abusive, she still loved you.
"yet" has a similar interpretation, but describes the state of a something, within a process, at any given time within a progression, up to that point in a process:
He is yet a child. -or- He is a child yet.
He's still a child.
In another usage, it is closer to "already." In questions, it asks whether something has happened "already" or up to the time of questioning, and additionally implies that the addressee is expected or required to do it:
Have you called your mother yet?
Here the speaker possibly commanded or just expects the addressee call his mother, or was informed of the his intention to call his mother, and is questioning whether he has done it. "Already" doesn't have this force - it merely questions the completion of the action logistically:
Have you already called your mother? I wanted to say hello to her. (=Am I too late?)
Note that the idiom "have yet to" is used with the above meaning to express that something expected still hasn't been done:
He has yet to call his mother. (= He's supposed to do it, but hasn't done it.)
I have yet to see that movie. (= I've been meaning to see it, but haven't gotten around to it.)
"Not yet" - or "yet" in any negative context - is the opposite of "already:"
My father already sent me a birthday present.
My father hasn't sent me a birthday present yet.
Finally, "yet" can function as a conjunction like "but, however" that emphasizes a result obtained or a situation that arises despite certain factor(s):
He's only been speaking English for a year, yet he sounds just like an
I've asked you a million times not to lock the door, yet every time I
come home, it's locked.
Note that "still" can replace "yet" here with the same meaning.