To sever something is to cut it off from the whole. If your girlfriend breaks up with you on your anniversary, you might respond by severing the blossoms off the roses you were planning to give her. (Just an idea.)
Sever rhymes with ever, but it looks like the word severe, which means "harsh." The similarity between sever and severe is a good reminder to reserve sever for harsh, unpleasant circumstances. If you’re trimming your fingernails, you might use the word clip, but sever wouldn’t be appropriate. If, however, you somehow cut off your finger while clipping your fingernails, you’re free to use the word sever — or any other word you like — on your way to the hospital.