See Humphreys, starting from II.5.3. Given a Coxeter system $(W, S)$ let $V$ be the free vector space on symbols $\alpha_s, s \in S$. We define a bilinear form on $V$ by extending

$$B(\alpha_s, \alpha_t) = - \cos \frac{\pi}{m(s, t)}$$

linearly, and we define a canonical linear representation $\sigma : W \to \text{GL}(V)$ respecting $B$ by extending

$$\sigma_s \lambda = \lambda - 2 B(\alpha_s, \lambda) \alpha_s$$

multiplicatively. This representation is faithful (II.5.4), so we can study $W$ using it. The most familiar case is when $B$ is positive-definite, and this occurs if and only if $W$ is finite (II.6.4). Cases where $B$ is positive-semidefinite include the affine Weyl group case, and in general what kind of geometry we get depends on the signature of $B$ (see for example II.6.8).

Alternately, the geometric pictures are perhaps clearest for Coxeter systems of rank $3$, where the corresponding groups are essentially the triangle groups. These can be understood as symmetries of tilings of the sphere, the Euclidean plane, or the hyperbolic plane by triangles depending on the signature of $B$. This picture should be compatible with the one above (I think one just needs to consider the action of $W$ on the "unit ball" in $V$) but I haven't checked the details.

The Geometry and Topology of Coxeter Groupsfor details. – HJRW Oct 25 '11 at 20:40