"[Previously], we saw that every closed oriented surface of genus g>1 has a hyperbolic structure. Actually, there is a lot of freedom in the construction, with the consequence that there is not just one hyperbolic structure, but many. ...

First, though, we must be more precise about what we mean by two hyperbolic structures being the same. There are in fact two important notions of equivalence, giving rise to two spaces: moduli space and Teichmuller space. Informally, in Teichmuller space, we pay attention not just to what metric a surface is wearing, but also to how it is worn. In moduli space, all surfaces wearing the same metric are equivalent. The importance of the distinction will be clear to anybody who, after putting a pajama suit on an infant, has found one leg to be twisted."

Bill Thurston in Three-Dimensional Geometry and Topology

Teichmuller space is a parameter space of marked metrics on surfaces of fixed topological type. (Note that a conformal class of metrics has exactly one constant-curvature hyperbolic representative, so we can think of the points in Teichmuller space as hyperbolic structures when convenient and equivalence classes of metrics when we prefer.)

That space of metrics itself has several metrics which give it its own geometry. (Meta-metrics?) The first (historically) is the Teichmuller metric, which is not Riemannian -- only Finsler -- but can be worked with very explicitly in terms of flat structures on the surface. A rival is the Weil-Petersson metric, which is Riemannian and geodesically convex (good) but incomplete (bad).

A lovely basic article on how to give "pairs of pants"-based coordinates on Teichmuller space:

Fenchel-Nielsen coordinates (by Kathy Paur for MIT undergrad journal of math)

For experts: a handful of useful articles for reference. Not a thorough selection; just the ones I've needed lately. (re-linked from the authors' web pages or from journal sites)