the fact that it may be a single gene may be debatable, as beast mentioned during the HoM series. "it's not just a single thing that makes a mutant a mutant, just like it's not just one thing that makes someone black." something to that accord. however, mutations occur when mutated genes, referred to as the "x" gene, are stimulated due to various stresses [most typically puberty.] special cases of irradiated parents birthing mutants whose powers were already active [beast, madrox] exist, but are atypical.
ultimate x-men is a better series to get into if you want to understand everything more easily. there're only 50-60 issues, i think, compared to approximately 700 in uncanny and adjectiveless x-men in the 616 line. ultimate changes many things about the regular universe, but also has many throwbacks to the 616 which fans of both universes can appreciate. the "true" x-men's much more topsy turvy than ultimates, though the last BIG event was onslaught back in '96. nothing's really been on that level since.
i also highly recommend UXN, considering their enormous archives and forum which you can use to find/ask for other information. and, i also highly warn against the movies, as they're horrible and i hope they die.