Your question is probably too general, I simply hope that you'd like to learn some personal experience of people who do their research in diophantine equations or who apply the equations to other areas.

Although my starting research (under- and postgraduate level) was the theory of transcendental numbers, quite tied to diophantine equations (especially the ones related to linear forms in logarithms), I did not try to work on these. (Maybe, because of my father's attempts to prove Fermat's Last Theorem.) But at one occasion I was "introduced" to the Erdős--Moser equation $1^k+2^k+\dots+(m−1)^k=m^k$ ($m$ and $k$ positive integers), Applications of pattern-free continued fractions, and was impressed by the beauty and power of the method which Leo Moser used in 1953 to show that no solution (except $1+2=3$) exists with $m\le 10^{10^6}$. (!) Moser's method could not give much more, and I was happy enough to collaborate on some new ideas and computational achievements of nowadays to significantly extend (after more than 50 years) Moser's bound.

Another favourite diophantine equation is Catalan's equation $x^p-y^q=1$ ($p,q>1$) which was rather recently solved by Preda Mihăilescu. He even managed to avoid linear forms in logs (which are far from beauty because of so many technicalities). There are many steps in the proof, treating some special cases, most of them using completely different methods of diophantine analysis and algebraic number theory. But they are just beautiful! For example, there are two different proofs of the nonsolvability of $x^2-y^q=1$ in integers $x>1$, $y>1$ for a prime $q>3$. The original one, due to Ko Chao ([On the diophantine equation $x^2=y^n+1$, $xy\ne0$, Sci. Sinica 14 (1965) 457--460]; also given in Mordell's "Diophantine equations"), uses the law of quadratic reciprocity in a very elegant way. Another one, due to E.Z. Chein [A note on the equation $x^2=y^q+1$, Proc. Amer. Math. Soc.
56 (1976) 83--84], is extremely short and elementary.

Summarizing, I would say that natural criteria for considering some diophantine equations (and ignoring other) are the simplicity of the equation (isn't $x^n+y^n=z^n$ simple?) and the beauty and novelty of methods to solve it. Probably, the usefullness of the equation has to be taken into account as well.