A Wedge is a small, hand held device used to store personalities that can be used as imprints on Dollhouse Actives. Roughly the shape and size of a personal computer hard drive, a single wedge is capable of holding enough information to form an entire personality. Personalities that can be stored on a wedge include an Active's original personality, or the personality build used for an Active's engagements. As a single wedge can only hold a single personality, multiple wedges are required for use by a Dollhouse facility, with backups of the original personalities kept away from the "primaries".
After his composite event, one of Alpha's first voluntary actions (as opposed to actions under the control of an imprint) was to destroy the wedges containing the personality of Karl William Kraft. (S1: "Omega") Characters have tried to liken the concept of a personality on a wedge to being a "soul", but they are also copyable and alterable. Other characters have suggested the "soul" is something else, independent of the personality/memories content of the brain; the show demonstrates that Echo intrinsically retains something of Caroline Farrell, much as Alpha is still part of his former self, Karl William Kraft, to the extent that Alpha retained some aspects of Kraft's personality even after he destroyed the wedge containing Kraft's own imprint (it has been speculated that the destruction of the Kraft wedge was motivated by Alpha's 'anger' at Kraft for 'abandoning' him to the Dollhouse). The philosophical implications of either interpretation are vast.
Wedges aren't very securely maintained. In "Omega", Alpha stole 38 wedges and easily destroyed a backup. The Dollhouse apparently does not keep off-site backups. In their defence, the whole place is already a top-secret, nearly-impenetrable underground facility, and the lack of internal defenses may simply be attributed to the assumption that nobody would be able to attack them in the first place.