Cannot Move Web Parts on Page

This simple little issue has cropped up at least twice in our environment. Most recently it happened because we started using a new alternate access mapping. Previously we encountered it when migrating from IE 6 to IE 7.

The Javascript needed for editing pages (and other stuff, no doubt) cannot run without adequate browser security. Check that the site is included in Trusted Sites or the Intranet Zone and that your policies (for the zone for the specific browser version) enable scripting.