In BrE only 'alternative' is correct (meaning one from a choice of two or more), and 'alternate' means something quite different: a zebra has alternate black and white stripes, that is two things follow each other, first one then the other, repeating. In AmE 'alternate' has both these meanings.
It's difficult to know how to be helpful without more information. What still confuses you that the other threads don't explain? Please give us a sentence in which you thing you might use these words, as an example.