Obviously, general supersymmetric sigma-models are sigma-models with the only requirement that they exhibit supersymmetry. A special subset of such models will also have other symmetries. In particular, models that also possess additional continuous symmetries unrelated to spacetime transformation (so-called 'internal' symmetries) are called gauged linear sigma models.
The supersymmetry in these sigma models provides several important restrictions. In particular, the ground states (states of lowest energy) in supersymmetric sigma models are easier to find than in non-supersymmetric theories. This is, in part, because they all have exactly zero energy and no excitations (particles) but also because they are invariant under supersymmetry - hence their name: supersymmetric vacua.
The entire Hilbert space (of states) in such models can be erected upon the supersymmetric vacua - one tower above every vacuum state. It turns out to be possible to perform certain simple transformations which either transform the Hilbert space of a model into itself or into the Hilbert space of another model. These are mirror and duality transforms, which often carry surprising and sometimes confounding implications for the geometry of the target space of the sigma model.
© Tristan Hübsch, 2008