How to make a Peyote Warped Square 

Julia Pretl animated the process of building a peyote-stitched Warped Square for the Contemporary Geometric Beadwork Open Source Library.
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You can choose to reinforce the center start with a few passes (or a knot, if that's your style) or you can wait to do that later when you finish. If you like your squares energetic, springy and cupped, by all means make a snug start. If you like them flatter, leave the start thread loose and tighten it to your liking later before weaving in.
Weave in the starting thread and your working thread after your square is the desired size, and consider reinforcing the edge with a final pass of thread through the last two rounds if you want it to be sturdy and stand up to play.
The "step-up" (moving from one finished round to the next) is beautifully illustrated here- pay attention to the final stitch of each round - to finish each round, you pass through two beads, which puts you in position to begin the next round.