Peyote Stitch is made by alternating connections.
First, string a row of beads.

For the second row and each row there after: Bring your thread from the end of the first row up through the first bead then bring the thread down through the second bead (skipping the bead underneath). Continue adding beads.

Peyote Stitch is often done in the round and, flat or round, is the most often seen stitch. As you add beads your project will automatically grown up and around. You can find this stitch used in pen and lighter covers for sale at PowWow's. It's also used in roaches and to cover the ends of feathers.

Peyote Stitch patterns are off centered. Much of the Native beading you find is done in Peyote Stitch.