In dynamical systems examples are common in which two or more attractors coexist and in such cases the basin boundary is non-empty. The purpose of this paper is to describe the structure and properties of basins and their boundaries for two-dimensional diffeomorphisms. If a two-dimensional basin has a basin cell (a trapping region whose boundary consists of pieces of the stable and unstable manifolds of a well-chosen periodic orbit), then the basin consists of a central body (the basin cell) and a finite number of channels attached to it and the basin boundary is fractal. We prove the following surprising property for certain fractal basin boundaries: a basin of attraction B has a basin cell if and only if every diverging path in basin B has the entire basin boundary \partial\bar{B} as its limit set. The latter property reflects a complete entangled basin and its boundary.