With the ones that work with NMM just click the download with NMM button and they will automatically download and be put into the folders they need to go in. Some mods you can't download manually, some you can't download with NMM. Now as for downloading mods, on the mod page (usually nexus) at the mods main page it will have buttons to download with NMM, download manually, or just 1 of those. Make sure its up and running before you start the game and play away.
So start NMM, make sure your on the current game your playing and thats basically it for 'using' it. On the top right there is a button arrow that you can switch between games depending on which game your playing. The first time you do it you can scan your computer for games that use NMM and it will locate them (like fallout 3, new vegas, skyrim, what ever).
To 'use' nexus mod manager just start the program. The catagories are only for you, if you want to keep track of putting them into different catagories so you know what you have, you don't need to put them into catagories for them to work its just a cosmetic thing for you as a player.