Star Trek is a franchise that can accommodate a large variety of concepts and still maintain its core identity, which is a real strength of the concept. Legally speaking, of course, any product with the Star Trek name produced by its license holders would qualify, but there are characteristics which serve to define anything, and that goes beyond legalities.
For Star Trek, I would say the most important factor is that it represents Roddenberry's vision of an optimistic future where humanity has grown to be better, not just as individuals, but as a species -- where we have learned to accept our differences and embrace the wonders life has to offer, always striving to improve ourselves and our knowledge of the universe.
I know some people feel that Deep Space Nine didn't live up to this vision, but I disagree. Yes, it dealt with darker material than Trek prior had (or more accurately dealt with material in a darker way), and it tested the heroes' principles, asking of them and us if those principles could be held on to in times of crisis, but it was hopeful for the future, and those heroes who got their hands dirty never became dirty themselves.
Of course, that's just my two cents worth. For what it's worth.
