the mapping class group
Pick up a surface and rearrange it with your hands without tearing — you can twist a handle, swap two holes, flip it over. Some such rearrangements can be slid continuously back to doing nothing (those are 'invisible'); others genuinely permute the surface's features and cannot be undone by sliding. The mapping class group collects exactly the genuinely different rearrangements: the symmetries of a surface up to continuous deformation. It is the surface's discrete symmetry group, and it is enormous and intricate even for a two-holed surface.
Precisely, for an oriented surface S the mapping class group MCG(S) (also written Mod(S)) is the group of isotopy classes of orientation-preserving self-homeomorphisms of S: take all homeomorphisms f: S -> S, and declare two of them the same when one can be continuously deformed (isotoped) into the other. Composition of maps gives the group operation. The Dehn twists — cut along a simple closed curve, rotate one side by a full turn, reglue — generate the whole group (Dehn-Lickorish theorem); finitely many of them suffice. For a genus-g closed surface MCG is a finitely presented, infinite group with rich algebraic structure.
Its central role is dynamical: MCG(S) acts on Teichmüller space T(S) properly discontinuously by changing the marking, and the quotient T(S) / MCG(S) is exactly moduli space M(S). So studying surfaces up to ALL symmetry means understanding this action. The Nielsen-Thurston classification sorts each mapping class into periodic, reducible, or pseudo-Anosov, and the pseudo-Anosov ones — which stretch one foliation and contract another by a fixed factor lambda > 1 — are the source of hyperbolic 3-manifolds fibering over the circle, tying surface dynamics directly to 3-manifold geometry.
On the torus T^2, the mapping class group is exactly SL(2, Z): a self-homeomorphism is determined up to isotopy by the integer matrix recording how it permutes the two basis loops of pi_1(T^2) = Z^2, and orientation-preserving forces determinant +1. A Dehn twist along one basis curve is the matrix [1, 1; 0, 1]; the map with matrix [2, 1; 1, 1] is pseudo-Anosov (it is Anosov, with stretch factor the larger eigenvalue (3 + sqrt 5)/2) and its mapping torus is a Sol-geometry 3-manifold.
Torus mapping class group = SL(2,Z); a pseudo-Anosov class has a Sol mapping torus, linking surface dynamics to 3-geometry.
The mapping class group is by convention the group of isotopy classes of orientation-PRESERVING homeomorphisms; including orientation-reversing ones gives the 'extended' mapping class group, of which MCG is an index-2 subgroup. Also, isotopy and homotopy of homeomorphisms agree for surfaces, but this is a theorem, not a definition.