Whether you need this or not depends on which mods you install, which should be clearly marked in the mod's requirements tab on the Nexus. Finally, some mods will require the Fallout 4 Script Extender, which is an alternate way of launching the game to enable more advanced scripting. If you have all the DLC, then the Unofficial Patch should also be a mandatory install. This, of course, is completely optional and unnecessary if you don't care about unlocking achievements. When playing with mods, achievements are normally disabled (because it's effectively cheating, I guess), so the Achievements mod simply re-enables achievements. The MCM adds an in-game option in the settings menu which will let you configure many mods individually from within the game itself not every mod has (or needs) MCM support, but it's great to have for the mods that do.

Next, there are few general mods you'll probably want to install right away, which are the Mod Configuration Menu and the Achievements mod.

Gopher's video guide for installing and using the Nexus Mod Manager is helpful for learning how to use it, and the mod manager is really sleek and intuitive once you're familiar with it. It also allows you to set your own mod load order (helpful for preventing mod conflicts), and enables virtual installs so that you're not directly replacing game files (less tampering with main game files means less chance of screwing something up and causing critical errors). From there, it's simply a matter of downloading and installing your desired mods - you can do this manually, typically by extracting files and folders and placing them directly in your game's main install folder, but I'd recommend using the Nexus Mod Manager because it makes keeping track of your mods much more convenient. It's pretty simple, really, and only involves editing the game's ini file.

If this is your first time modding Fallout 4, note that you'll have to tweak some settings to enable unofficial mods (basically anything that you aren't paying for directly through Bethesda's in-game store).

Fallout 4 mod review