What does it mean for something to be a model of hyperbolic space?

In the book "Non-Euclidean Geometry and Curvature" by James W. Cannon, the author uses the term "analytic models of hyperbolic space." (p. 19) Some examples are the Klein model and the Hyperboloid model, which are mentioned on Wikipedia as well. However, he does not explain what such a model is. Why do they qualify as models of hyperbolic space, and how can both be valid?

You are right - a model of hyperbolic space is a mathematical structure in which we define "points" and "lines" so that the modified Euclidean postulates (with the parallel postulate replaced by its hyperbolic equivalent) are true.

The 3 most common models of 2D hyperbolic space are the hyperboloid model, the Klein model and the Poincare model. In each model "points" are still normal geometric points, but "lines" are defined as geodesics in a non-Euclidean metric.

In the hyperboloid model the "lines" all lie on one sheet of a hyperboloid. In the Klein model "lines" are Euclidean lines on a plane but distance is no-Euclidean, so points at infinity lie on the circumference of a circle. In the Poincare model "lines" are arcs of circles on a plane; again, points at infinity lie on the circumference of a circle.

If you embed each of these 2D models in 3D Euclidean space then they can be related by very interesting and beautiful projections.

A model of a theory is a concrete, constructed example where the theory is applicable because the axioms are verified.

For example $\mathbb{Z}$, $\mathbb{C}$ and $S_6$ are all groups, meaning they are models for the group theory. They satisfy the group axioms and and you can apply group theory results to them. They are also very different from each other.