Still refers to an action or situation that hasn’t finished, or that is ongoing, up to and including the present (or the time mentioned). Often this "not being finished" is surprising, going against expectations.
Examples:
I’m still writing the report and the deadline is in five minutes!
Six o’clock and you’re still in the office?
I went back two years later and they still hadn’t finished. Unbelievable!
Yet is mainly used in questions and in negative sentences, and concerns something which we expect to happen.
Not yet is used to say that something we expect to happen is still in the future.
Examples:
Have the documents arrived? - Not yet.
The visitors from the States haven’t arrived yet.
Note: We can also use "still" instead of "yet" in the above example by placing it before the be-verb "haven't":
The visitors from the States still haven't arrived.
In questions, we use yet to ask whether something expected has happened.