I ran across Picard categories in a totally different area of mathematics, but maybe it helps.
In short, a Picard category is a group object in the category of groupoids.
Picard categories come up when you study Picard stacks. Roughly, a Picard stack is a sheaf of Picard categories. The classical example is taking a two-term (perfect??) complex of sheaves, and asssociating to such a complex the Groupoid quotient of the one term by the other. This is important when you want to produce a geomteric object from such a complex. This is an important tool in defining virtual fundamental classes as in http://arxiv.org/abs/alg-geom/9601010.
Before I tell you to many things that are not true, here are the references I know of:
Lecture notes of Martin Olsson at http://math.berkeley.edu/~molsson/MSRISummer07.pdf, Lecture 5. You can even watch that on video: http://www.msri.org/calendar/sgw/WorkshopInfo/419/show_sgw
The definitive reference is Expos´e XVIII of SGA 4. You used to be able to get that at the Grothendieck circle. Now its still availabe on all the usual Russian sites.
And finally there are very friendly and down to earth lectures of Barbara Fanechi Fantechi at http://www.openeya.org/sissa/ . I think lecture 3 or lecture 4 is about Picard categories.