The resolution of which conjecture/problem would advance Mathematics the most? 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.
 A: I conjecture the question to be premature ...
I would nominate the Riemann Hypothesis, since it is clear that something occurs that we fundamentally don't understand. But folding other things in with RH, such as the Artin Conjecture, is known to be a good idea (since Weil). And the good behaviour of L-functions should be expressed by a "geometric" principle. Such a thing, if convincingly formulated, would have a strong claim.
There is presumably another major conjecture to do with how K-theory would rule geometry. Algebra rules, subordinating geometry then analysis. Such a clutch of conjectures would delineate the reach of structure, at least into the heart of the mathematical heritage of the 19th century. 
Anyway, top-down questions provoke top-down answers, by suggesting "incremental" progress is beside the point. But it isn't, of course.
A: I am going to submit the Baum-Connes conjecture because probably nobody else will and I believe its importance is quite understated.  The conjecture asserts that the assembly map from the K-homology of the appropriate classifying space of a group $G$ to the K-theory of the reduced group C*-algebra of $G$ is an isomorphism for every group $G$.  A proof would immediately imply two old conjectures in completely different areas of mathematics:


*

*Injectivity implies the Novikov conjecture in high-dimensional topology

*Surjectivity implies the Kadison-Kaplansky conjecture in analysis


Aside from that, the conjecture is deeply linked to a variety of other open problems in differential geometry and topology, for instance in the theory of positive scalar curvature invariants and eta invariants.  A bit more speculatively, the conjecture suggests a number of deep dualities in representation theory which could shed light on open problems in that area as well; see, for instance, Aubert-Baum-Plyman's recent work on local Langlands.
Finally, unlike with many conjectures in mathematics, a counter-example would in some ways be more exciting than a proof.  Some experts think that $SL(\mathbb{Z},3)$ is a counter-example to surjectivity, but proving it is well beyond the reach of current techniques.  Counter-examples to injectivity might require completely new ways to construct discrete groups, and would undoubtedly have applications to geometric group theory.
A: How about the Millennium Prize Problems
