The construction for a regular hexagon is a favorite of my students every year and the one most of them choose as a basis for their geometric design project. We are simply going to begin with a line and use our compass to draw three congruent circles along it. Start with the center circle. Mark the two points where this circle crosses your line, and use them as center points for two more matching circles. On your first circle, you now have six evenly spaced points, which can be connected to form your hexagon. Connecting points in different combinations will form many other polygons as well.
The images below are examples of designs from my sixth grade students that use this construction as a starting point.