  This article deals with nonwandering (e.g. area-preserving) homeomorphisms of
the torus $\mathbb{T}^2$ which are homotopic to the identity and strictly
toral, in the sense that they exhibit dynamical properties that are not present
in homeomorphisms of the annulus or the plane. This includes all homeomorphisms
which have a rotation set with nonempty interior. We define two types of
points: inessential and essential. The set of inessential points $ine(f)$ is
shown to be a disjoint union of periodic topological disks ("elliptic
islands"), while the set of essential points $ess(f)$ is an essential
continuum, with typically rich dynamics (the "chaotic region"). This
generalizes and improves a similar description by J\"ager. The key result is
boundedness of these "elliptic islands", which allows, among other things, to
obtain sharp (uniform) bounds of the diffusion rates. We also show that the
dynamics in $ess(f)$ is as rich as in $\mathbb{T}^2$ from the rotational
viewpoint, and we obtain results relating the existence of large invariant
topological disks to the abundance of fixed points.
