Task
Sometimes when attempting to move web-parts within SharePoint they will just not move. There is a very simple solution to this issue using the web-part properties.
NOTE: This works on Web Part pages, not on Wiki Pages.
Solution
Edit Web-Part
Edit the Page, click on the arrow in the top right of the web-part and click Edit Web Part.
The two areas we are going to look at are under the Layout section:
Zone: This allows you to transfer the Web Part to a different Web Part Zone.
Zone Index: This is the ordering of the Web Parts in the Zone
So to move the Web Part, just modify these values and the Web Part will adjust its location without attempting to use the Drag & Drop feature that rarely works properly.