I'd like to build on Graham and anonymous' answers. There are lots of ways that a free resolution can be used to obtain interesting geometric information, but I think of it in a rather different way than the "untwisting" suggested in your question. This is most easy to discuss in the case that you are studying a projective scheme $X\subseteq \mathbb P^n$. Let $S_X$ be the homogeneous coordinate ring $S_X$ of $X$, considered as a graded $k[x_0, \dots, x_n]$-module. Then, you can define the minimal free resolution of $S_X$:

$
0\to F_{n+1}\to \dots \to F_1\to F_0\to S_X \to 0
$

where each $F_i$ is a free, graded module over the polynomial ring.

As Graham points out, the maps in the free resolution carry a ton of information. But since we are in the graded case, even if you forget about the maps, you still obtain a lot of information. For instance, if you only know the graded ranks of each $F_i$, then you can recover the Hilbert function of $X$, as well as some other interesting numerical invariants: Hilbert polynomial, Castelnuovo-Mumford regularity, depth. You can also determine if $S_X$ is a Cohen-Macaulay or Gorenstein ring, simply from the graded ranks of the $F_i$.

Those are all straightforward applications: but there is also a large body of literature on how to extract more subtle geometric information about $X$ from the ranks of the $F_i$. This is essentially the focus of Eisenbud's "The Geometry of Syzygies," and that book is full of surprising examples.

I think that one example can be quite enlightening, so I'll yank Theorem 2.8 from Eisenbud's book. This says that if $X$ is the union of $7$ points in linearly general position in $\mathbb P^3$, then these $7$ points lie on a twisted cubic curve if and only if $\text{rank}(F_1)>4$. In other words: in this case, the rank of $F_1$ measures a subtle geometric property of the $7$ points (i.e. whether or not they lie on a twisted cubic).