This can mean several different things. See nLab for one of them. Exact category: Means one thing in Borceaux vol 2 and another thing inn Quillen.
http://ncatlab.org/nlab/show/Quillen+exact+category
For an expose of the Quillen stuff, see Buehler
In the 50s, this was defined by Buchsbaum as “an abelian category but without the requirement that direct sums exist”. Is this also the modern definition?
LNM0236 treats exact and regular categories, and “categories of sheaves”, but doesn’t look terribly interesting at first sight
nLab page on Exact category