This is an impossibly broad question, and makes the unwarranted assumption that Mathematics is a uniform field. It might make more sense to ask the same question restricted to, say, Mathematical Logic, or Differential Geometry, or Number Theory. Nevertheless, I think there may be a sense in which certain fields of mathematics are (temporarily) blocked by major unresolved conjectures, and others are not. In which case, a breakthrough in a "blocked" field might advance Mathematics—broadly construed—the most.
For example, prior to 2002, I would have ventured: the Poincaré conjecture & Thurston's geometrization conjecture. But since Perelman's resolution, I see (in my limited vision) no equivalent major conjecture outstanding in Differential Geometry. Similarly, one might might view Fermat's last theorem as a key blockage within Number Theory before the 1995 Wiles-Taylor resolution; or twin primes before Zhang's 2013 breakthrough and subsequent polymath's improvements.
My parochial, biased viewpoint is that resolution of the P=NP question, tomorrow, would represent the greatest advance in Mathematics. But perhaps others think that, e.g., resolution of the Riemann hypothesis would represent the larger advance? Of course, how a question is resolved makes a huge difference: A narrow proof is less an advance than a broad reconceptualization that reshapes the landscape.
Q. I welcome responses specific to narrow fields, but also remarks on Mathematics (if such exists!) as a whole—from which we may all learn from one another.
Editor's note: there is a meta thread about this post at Should we reopen "The resolution of which conjecture/problem would-advance mathematics the most?". Please read that before casting open/close votes.