x^2 - ny^2 = 1 is interesting for at least two reasons:  on the one hand, x^2 - ny^2 is a norm from the quadratic field, so the equation has to do with the rather natural question of studying units in real quadratic fields.  On the other hand (or, really, on a different finger of the same hand) it is just what you want to study if you are interested in rational approximations to square roots of integers, which in some sense are the "simplest" irrational numbers and thus the first context in which you might think about approximating irrationals by rationals.

Similarly, the Fermat and generalized Fermat equations are quite natural in the following sense:  there is a long history of studying the interplay between addition and multiplication in integers, and in particular the additive relations between multiplicatively defined sets (primes, perfect powers, etc.)  In this context it makes sense to think about x^n + y^n = z^n and things like the Goldbach conjecture.  What makes the former more natural?  In some sense, it is natural because there's an approach to it!  It turns out that the equation x^n + y^n = z^n is intimately related to the geometry of P^1 - three points (in some sense the algebraic curve on which all others are based) and to the closely related object X(1), the moduli space of elliptic curves.

There is no hard and fast rule for "which Diophantine questions are interesting" -- but in general it is not so far off to say that the ones which are interesting are the ones where we have at least some idea how to attack them, because the reason we have some idea how to attack them is typically because they're connected to some other mathematical objects of interest.