There's at least the following relationship: both the Brouwer fixed-point theorem and the hairy ball theorem are easy consequences of a more-highbrow thing called the Lefschetz fixed-point theorem.
Unfortunately even the statement of the Lefschetz fixed-point theorem is a bit complicated, but let's see what I can do. I'll have to miss out most of the details. Depending on how much mathematics you know, it may not make much sense. But here goes.
If you have a topological space X, there are a bunch of things called its "homology groups": H_0(X), H_1(X), H_2(X), and so on. I will not try to define them here. If you have a continuous map f from the space X to the space Y, then it gives rise to corresponding maps from H_k(X) to H_k(Y).
The machinery that manufactures homology groups can be parameterized in a certain way so that you can get, instead of the ordinary homology groups, "the homology groups over the rational numbers", "... over the real numbers", and so on. (These can actually be obtained fairly straightforwardly from the ordinary homology groups "over the integers".) If you do it "over the rational numbers" or "over the real numbers" then the resulting things are actually _vector spaces_, and if your space is reasonably nice they're _finite-dimensional vector spaces_.
(What's a vector space? Well, there's a formal definition which is great if you're a mathematician. If not: let n be a positive integer; consider lists of n numbers; for any given n, all these lists collectively form a "vector space of dimension n". You can do things like adding two lists (element by element) or scaling the values in a list by any number (just multiply them all by the number). A finite-dimensional vector space is a thing where you can do those operations, that behaves exactly like the lists of n numbers, for some choice of n.)
And then the maps between these vector spaces, that arise (magically; I haven't told you how) out of continuous functions between topological spaces, are linear maps. You can represent them by matrices, with composition of maps (do this, then do that) turning into multiplication of matrices.
OK. Now I can kinda-sorta state the Lefschetz fixed-point theorem.
Suppose X is a compact topological space, and f is a continuous mapping from X to itself. Then you get corresponding maps from H_k(X) to itself, for each k. For each of these maps, look at the corresponding matrix, and compute its trace: the sum of its diagonal elements. Call this t_k. And now compute t_0 - t_1 + t_2 - t_3 + ... . (It turns out that only finitely many of these terms can be nonzero, so the sum does make sense.) Then: If this is not zero, then f must have a fixed point.
So, whatever does this have to do with the Brouwer fixed-point theorem or the hairy ball theorem?
The Brouwer fixed-point theorem is about maps from the n-dimensional ball to itself. It turns out that all the homology groups of the n-dimensional ball are trivial (have only one element) apart from H_0, and that whatever f is the map from H_0 to itself that arises from f is the identity. And this turns out to mean that the alternating sum above is 1 - 0 + 0 - 0 + ... = 1. Which is not zero. So the map has a fixed point.
The hairy ball theorem says that a continuous vector field on the 2-dimensional sphere has to be zero somewhere. Suppose you have a counterexample to this. Then you can make a whole family of maps from the 2-dimensional sphere to itself, each of which looks like "start at x and move a distance epsilon in the direction of the vector at x". If epsilon=0 then this is the identity map. If epsilon is positive and sufficiently small, then the fact that the vector field is never 0 guarantees that the map does actually move every point; in other words, that it has no fixed points.
But all the terms in that infinite sum that appears in the Lefschetz fixed-point theorem are (so to speak) continuous functions of f. And it's not hard to show that the value of the sum for f = identity is exactly 2. So for very small epsilon, the value of the sum must be close to 2, and in particular must be nonzero. So, for small enough epsilon, we have a map with no fixed points and a nonzero value of the sum, which is exactly what Lefschetz says can't happen.