To a space (typically with singularities) of certain kind (there are variants) one associates category whose objects are called perverse sheaves. They are neither perverse nor sheaves; but they are related to some sheaf categories and notably generalize the intersection cohomology. Perversity is there a parameter involved in grading of intersection cohomology groups. Another feature similar to sheaves is that they are somehow determined by the local data; there is a famous gluing due Sasha Beilinson. In one of the approaches (see MacPherson’s notes) there are even modified Steenrod-Eilenberg axioms stated for intersection cohomology.
A. A. Beilinson, J. Bernstein, P. Deligne, Faisceaux pervers, Astérisque 100 (1980) MR86g:32015
scan of old notes from MacPherson pdf
Reinhardt Kiehl, Rainer Weissauer, Weil conjectures, perverse sheaves and l’adic Fourier transform, Ergebnisse Der Mathematik Und Ihrer Grenzgebiete 42, Springer 2001.
Ryan Reich, Notes on Beilinson’s “How to glue perverse sheaves”, arxiv/1002.1686