There are two possible ways:
1. If you want to re-arrange many products, the best way is to import them with new parameters.
2. The other way is as follows:
go to the Catalog → Categories page
find the necessary category in the left panel, click it, then open the "Category products" tab.
press the "Remove all" button; the products will be removed from this category, however the products themselves will not be removed.
then go to the "Category products" tab of the destination category, press "Add products" and add all necessary products at once there.