The best way to move Albums and Categories is using the Organizer. The Organizer is accessible through the main menu "Organize" >> "Organizer" (1a)

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or the "Organize" option (1b) in the various "Tools" menus like Albums, Category, or "Organize Photos" in the "Tools" for Photos. Its workspace and range of uses are summarized in the Organizer - general article.

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Once in the Organizer, green tick or red cross mark can help you identify valid / invalid operations. For instance, moving an album into a category would work (the green tick mark is circled) but a category cannot be moved to an album :

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How do I move an album from my homepage to a different category?  There are two ways :

(A)
Firstly, you find the album in the navigation panel (left-side panel displaying albums, categories and featured albums & featured photos), catch the album with your mouse (click on it and hold it using the left mouse/touchpad button), drag it into the destination category and once you see a green tick, you can release/drop it. The operation is illustrated with an album named "First try" being moved to the "nature" category (2).

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(B) The second way is similar to the way photos are moved. 
Once you see the chosen album in your contents panel (right-side panel displaying contents of your homepage, categories or albums), you catch it (2), a blank space will appear while you drag the album to the destination category on the left (3). Photos behave similarly (for more see Organizer - Copying Photos).

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After both (A) or (B) method, you will be notified about the operation being performed (4). That should not take longer than few minutes depending on the number of items. You may use the time to upload photos, close the Organizer or do almost anything, unless the task is directly related to the items being moved.

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The Organizer also allows ordering/arranging of categories/albums/photos, and other options including the management of the featured albums and photos. Furthermore, it allows multiple photo/video editing, downloading of the uploaded originals, and applying watermarks.

Please note :  the Organizer does not allow you to drag any item from the navigation panel to the contents panel nor from the contents panel into a different item in the contents panel. It is also impossible to move categories and albums at once.