From the perspective of algebraic geometry, Teichmüller theory is an analytic approach to moduli spaces of curves. To keep things simple, let $g\geqq 2$ be an integer and consider the moduli space $\mathscr{M}_g$ of smooth and complete algebraic curves of genus $g$. This is a fairly complicated object: a Deligne-Mumford stack over the integers, nothing that is easy to describe in any way. The associated complex analytic "space" $\mathscr{M}_g(\mathbf{C})$ is still something with a weird structure: a complex orbifold which is not a manifold. But its universal covering space, which can be identified with the Teichmüller space $\mathscr{T}_g$, is a true complex manifold. It has several nice properties as compared with $\mathscr{M}_g$:

• Teichmüller space is biholomorphic to an open domain in $\mathbf{C}^{3g-3}$. This is Bers' embedding theorem.
• Teichmüller space is diffeomorphic (forgetting the complex structure) to $\mathbf{R}^{6g-6}$, and there is a very intuitive system of coordinates, called Fenchel-Nielsen coordinates, realising such a diffeomorphism. On the other hand, even when you forget the stack structure, $\mathscr{M}_g$ is a variety of general type for $g\ge g\geqq 23$, which means that you can only embed it in projective space $\mathbf{P}^d$ where $d$ is "much" larger than the dimension of $\mathscr{M}_g$, and you need "many" equations to cut out its image. So there is no "economical" algebraic coordinate system on $\mathscr{M}_g$ in general.
• Complex geodesics in $\mathscr{T}_g$ for a natural metric, the Teichmüller metric, give families of algebraic curves which have a nice and intuitive geometric description, called Teichmüller disks. In quite a few cases they descend to algebraically defined curves in $\mathscr{M}_g$ which are consequently called Teichmüller curves. One can say much more about their geometric and number-theoretical properties than for general curves in $\mathscr{M}_g$. They form an active area of research in these years.

Another application of Teichmüller theory to moduli spaces of curves is that it gives rise to an isomorphism between the mapping class group $\Gamma_g$ of a closed oriented surface of genus $g$ and the (orbifold) fundamental group of the moduli space $\mathscr{M}_g(\mathbf{C})$, so it provides a link between the topology of moduli spaces and the topology of surfaces.

From the perspective of algebraic geometry, Teichmüller theory is an analytic approach to moduli spaces of curves. To keep things simple, let $g\geqq 2$ be an integer and consider the moduli space $\mathscr{M}_g$ of smooth and complete algebraic curves of genus $g$. This is a fairly complicated object: a Deligne-Mumford stack over the integers, nothing that is easy to describe in any way. The associated complex analytic "space" $\mathscr{M}_g(\mathbf{C})$ is still something with a weird structure: a complex orbifold which is not a manifold. But its universal covering space, which can be identified with the Teichmüller space $\mathscr{T}_g$, is a true complex manifold. It has several nice properties as compared with $\mathscr{M}_g$:
• Teichmüller space is biholomorphic to an open domain in $\mathbf{C}^{3g-3}$. This is Bers' embedding theorem.
• Teichmüller space is diffeomorphic (forgetting the complex structure) to $\mathbf{R}^{6g-6}$, and there is a very intuitive system of coordinates, called Fenchel-Nielsen coordinates, realising such a diffeomorphism. On the other hand, even when you forget the stack structure, $\mathscr{M}_g$ is a variety of general type for $g\ge 23$, which means that you can only embed it in projective space $\mathbf{P}^d$ where $d$ is "much" larger than the dimension of $\mathscr{M}_g$, and you need "many" equations to cut out its image. So there is no "economical" algebraic coordinate system on $\mathscr{M}_g$ in general.
• Complex geodesics in $\mathscr{T}_g$ for a natural metric, the Teichmüller metric, give families of algebraic curves which have a nice and intuitive geometric description, called Teichmüller disks. In quite a few cases they descend to algebraically defined curves in $\mathscr{M}_g$ which are consequently called Teichmüller curves. One can say much more about their geometric and number-theoretical properties than for general curves in $\mathscr{M}_g$. They form an active area of research in these years.
Another application of Teichmüller theory to moduli spaces of curves is that it gives rise to an isomorphism between the mapping class group $\Gamma_g$ of a closed oriented surface of genus $g$ and the fundamental group of the moduli space $\mathscr{M}_g(\mathbf{C})$, so it provides a link between the topology of moduli spaces and the topology of surfaces.