Norman "Shiv" Logan was an inmate of the Bayside Correctional Facility.
Logan presumably acquired his nickname for his use of a homemade knife. It is unclear what crime he had been convicted of, or how long his sentence was, but he quickly acquired a reputation for being stupid. This made him a perfect pawn for prison guard Daniel Reese's plot to frame an inmate for the murder of Reese's lover, Lindsey Bishop.
Reese approached Shiv in secret, offering to help him escape (whatever reason he had for doing so, Shiv didn't seem to have asked). He gave Shiv a chisel, and instructed him exactly how to dig his escape tunnel, and when to go down it.
On the day of the planned escape, however, Jack Monk Jr. was placed in Shiv's cell. When Shiv uncovered his escape tunnel, Jack knocked him unconscious and crawled down it himself - unwittingly becoming Reese's patsy.
To cover his tracks, Reese killed Shiv - ironically, by stabbing him with a homemade shiv.