Per my Step By Step SharePoint 2010 book, to move a web part (Announcements Web Part) to a different place on my share point page:
Click site actions
Click edit page
Move mouse over title bar of web part (you are moving) until pointer changes to a hand and while holding down mouse button drag the web part to a different spot on your Share Point page and release mouse. A normal drag and drop, right ?
Well, my Announcements Web Part does not move anywhere, just stays in the same spot that Share Point put it when I added it to Share Point page.