Can I remove a photo from an album/folder, but not delete it from my photo library?

I am trying to organize my photos so that every photo is in an album. ( I want to be able to see individual photos that aren't in albums, so I can add them to the correct album.....so there are NO strays.)

First I made one GIANT folder of "ALL PHOTOS." As I assign a photo to an album/folder, I want it to be removed from the "ALL PHOTOS" folder I made. When I try "deleting" it from this folder, it deletes it, entirely. As assign a photo to a designated album/folder, I want to know how to remove it from the "ALL PHOTOS" folder - so it appears in only one folder. Is this possible?