When I was a child, we used to pick up geodes out of creek beds. Exactly what are they made up of, and how do they come about?

Geodes begin as bubbles in volcanic rock or as animal burrows, tree roots or mud balls in sedimentary rock. Over time, the outer shell of the spherical shape hardens, and water containing silica precipitation forms on the inside walls of the hollow cavity within the geode. The silica precipitation can contain any variety of dissolved minerals, the most common being quartz, but amethyst and calcite are also found.

Over a period of thousands of years, layers of silica cool, forming crystals of different minerals within the cavity. Different types of silica cool at varying temperatures, thus creating layers of different types of mineral crystals.

Good answer, but I would take exception to the descibing the crytalization process of geode formation as involving "cooling". The solutions that contain the minerals that are preciptated out to form the geode are derived from either ground-water or in marine environment the sea and are already cool. A more reasonable explaination to my mind would be that the layering of different minerals is caused by the most highly concentrated mineral crystalizing first with the rarer components of the solute taking longer to form crystals.

See:

Desert USA - Geodes
University of Michigan - Geodes
Australian Museum - Concretions, Thunder Eggs and Geodes