"[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 spaceandTeichmuller 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__

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)

- Linch's thesis: comparisons between Teich and WP metrics
- Masur's thesis: Teich metric not CAT(0)
- Masur developing boundary theory of Teichmuller space
- Brock survey on pants and the WP metric
- Masur-Wolf: Teich metric not Gromov hyperbolic
- Kaimanovich-Masur I: boundary of the mapping class group
- Kaimanovich-Masur II: boundary of Teich
- Masur-Minsky I: hyperbolicity of the complex of curves
- Masur-Minsky II: hierarchies