A cool application which I can somehow appreciate is Van den Bergh's proof of dimension $3$ case of the Bondal-Orlov conjecture that two birational smooth Calabi-Yau varieties $X,X'$ have equivalent derived category $D(X) \sim D(X')$. Note that since one can construct the pluricanonical ring from $D(X)$ , this is a generalization of the Batyrev's conjecture that they have the same Hodge numbers. The dimension $3$ case was first proved by Bridgeland, but Van den Bergh's proof uses non-commutative stuff in a very concrete way. Some references can be found in this question I asked.
It goes as follows: by some mimimal model program results, in dimension $3$ any birational $X,X'$ are related by a series of flops $X \to Y \leftarrow X^+$. So we only need to prove $D(X) \sim D(X^+)$. Then one builds a special vector bundle (sum of an exceptional collection of line bundles in this case, some perverse stuff!) on X. Pushforward to $Y$, one gets a coherent sheaf $E$. Let $A= End(E)$. The funny thing is that $D(X) \sim D(A)$. Note that $A$ is non-commutative.
Now, do the same thing for $X+$ one gets $A+$, say. But it is fairly easy to prove that $D(A) \sim D(A^+)$ directly on $Y$, so we are done. If we carry out this on an Atiyah's flop or Reid's pagoda, one can see actually that $A$ and $A+$ are the same. This indicates that the non-commutative route can simplify things.
There seem to be many experts on this site, so surely you will get a lot of much better answers. But this example seems most down to earth for me, and the conjectures come from complex geometry (motivated by physics?) so I hope it helps you as well.
EDIT: here is something to complement Lieven's great answer above: given $A$ one can actually construct back $X$ as a moduli space of certain A-representations (see Section 6 of this). One needs $A$ to have finite global dimension so $X$ can be smooth (the fact that $X$ is smooth is proved via the very algebraic intersection theorem). In dimension $3$, this example explains Lieven's sentence:
further, what a crepant resolution of a quotient singularity is to you, is to NCAGers the moduli space of certain representations of a nice noncommutative algebra over the singularity.