Aside from his work on the foundations of mathematics, which others have already elaborated on, earlier in his career Voevodsky also proved the Milnor conjecture in algebraic geometry. The Milnor conjecture relates Milnor K-theory to Galois cohomology (or, equivalently, etale cohomology, since we are dealing with the case of a field). It is related to Hilbert's Theorem 90, which has historical origins in number theory (going all the way back to Ernst Kummer's work in 1855).

Namely, in its modern formulation, Hilbert's Theorem 90 says that $H^{1}(G(k_{s}|k);k_{s}^{*})$ is trivial. By the Kummer exact sequence, we then get the result that

$k^{*}/2\cong H^{1}(G(k_{s}|k);\mathbb{Z}/2)$.

Milnor's conjecture states that

$K_{n}^{M}(k)/2\cong H^{n}(G(k_{s}|k);\mathbb{Z}/2)$

for $k$ of characteristic not equal to $2$, where $K_{n}^{M}(k)$ is the Milnor K-theory, defined as the quotient of the tensor algebra of $k^{*}$ by the ideal generated by $a\otimes(1-a)$ for $a\in k^{*}-\{1\}$. For $n=1$ we have $K_{1}^{M}(k)=k^{*}$, and the statement of the Milnor conjecture is the same as the result given above. We can therefore think of the Milnor conjecture as its generalization to higher $n$.

As a side comment, I'll mention that Milnor K-theory may be viewed as an "approximation" to algebraic K-theory, and agrees with Quillen's later definition of algebraic K-theory for $n\leq 2$. Algebraic K-theory is very much related to ideas in homotopy theory in algebraic topology.

A survey of Voevodsky's proof can be found here:

Voevodsky's Proof of Milnor's Conjecture by Fabien Morel

Later on Voevodsky also worked on the more general version of the Milnor conjecture, the Bloch-Kato conjecture, also now known as the norm residue isomorphism theorem. Material on the Bloch-Kato conjecture may be found on Charles Weibel's website:

Charles Weibel's Home Page

In the course of his work on the Milnor conjecture and the Bloch-Kato conjecture, Voevodsky developed motivic homotopy theory. This is a version of homotopy theory for algebraic geometry, and in order to transplant this theory from algebraic topology certain technical obstacles must first be surmounted, such as the unit interval not being an algebraic variety, and so it must be replaced by the affine line $\mathbb{A}^{1}$. A nice introduction to these ideas may be found in the book "Motivic Homotopy Theory" by Dundas, Levine, Rondigs, Ostvaer, and Voevodsky himself.

Voevodsky also developed, for the same purpose, a triangulated category of "mixed motives". Mixed motives are a generalization of pure motives, which is related to the search for a "universal cohomology theory". An introduction to the theory of motives, which however stops just short of mixed motives, is here (I'm linking to the top level of the site, as per the author's request, so just look for the article in the sidebar):

Motives - Grothendieck's Dream by James S. Milne

It is worth noting that a theory of mixed motives does not exist at the moment - the triangulated category of mixed motives is kind of the "next best thing". Other approaches were made by Huber-Klawitter, Hanamura, and Levine. A more thorough discussion of these ideas can be found in the book Une Introduction aux Motifs by Yves Andre, or, for an English reference, there's also a brief discussion in the book Feynman Motives by Matilde Marcolli, or the following book available online:

Noncommutative Geometry, Quantum Fields, and Motives by Alain Connes and Matilde Marcolli

Finally, this is much more elementary, but I've found the following popular exposition (on video) by Voevodsky to be an abstract, yet intuitive, and ultimately elegant presentation of homotopy theory:

An Intuitive Introduction to Motivic Homotopy Theory by Vladimir Voevodsky

There's also a pdf transcript of that talk.

The Work of Vladimir Voevodsky. Proceedings of the International Congress of Mathematicians 2002. $\endgroup$ – Peter Heinig Oct 1 '17 at 4:33