co.combinatorics - Intersection homology for toric varieties

There is a simple and beautiful description in terms of commutative algebra (repeatedly calculating global sections and taking a projective cover). The work of Braden-MacPherson cited by Alexander is relevant, but only for certain toric varieties (those admitting affine pavings). Also, the Braden-MacPherson paper is really aimed at handling the case of flag varieties etc., which is more complicated than toric varieties.

I think the first combinatorial description was given by Bernstein and Lunts at the end of their book on equivariant sheaves:

Bernstein, Joseph; Lunts, Valery
Equivariant sheaves and functors. LNM 1578. Berlin: Springer-Verlag.

This was then abstracted to arbitrary (perhaps non-rational) polytopes here:

Bressler, Paul and Lunts, Valery, Intersection Cohomology on Nonrational Polytopes,
Compositio Mathematica, Volume 135, Issue 3, pp 245-278.
http://arxiv.org/abs/math/0002006

There is parallel work by BBFK:

Gottfried Barthel, Jean-Paul Brasselet, Karl-Heinz Fieseler, and Ludger Kaup
Combinatorial intersection cohomology for fans, Tohoku Math. J. (2) Volume 54, Number 1 (2002), 1-41.

All of this is summarized quite nicely in Kirwan and Wolf, An introduction to Intersection Cohomology Theory, Second Edition, Chapman and Hall, 2006.