Addon
Addons or mods are modifications to the official version of the game. They either add new parts, systems and mechanics to the game or replace existing ones. Addons are generally made by the community rather than the developers and can greatly enhance your playing experience. Due to their widespread popularity, Squad offers an official website for exchanging them, the Kerbal SpacePort.
How to install addons
Provided you downloaded an addon from the SpacePort, you should first unpack the .zip archive and then copy the unpacked contents to the appropriate folder in your KSP directory. Here's how.
Open up your KSP directory and go to the "GameData" folder. In another window, find the (unpacked) contents of the .zip archive that you just extracted. As of update 0.20, compatible addons should just show a [mod name here] folder. Copy that folder to the GameData folder, and you're done! All you have to do from there is start the game.
Uninstalling is just as simple as deleting the folder with the name of the addon that you want to remove. However, any ships created with those addons will not load next time you start your game, once uninstalled.
Directory system
Note: as of the 0.20 update, all of the addon folders are contained in a parent folder (containing the name of the mod downloaded) in the GameData folder. For "legacy" (pre-0.20) addons, the folders were at the root of the .zip file - requiring the user to move the "Parts" and/or "Plugins" folders to the corresponding KSP directory folders. With many mods installed, it may have been hard to tell what parts were mods and what parts weren't when uninstalling. Now, each addon gets its own folder in the KSP directory (instead of sharing one with the game itself), simplifying things.
- Inside of the .zip file will be the folders "Parts", "Plugins", and (occasionally) "Source". There may also be documentation, usually in the form of a "README" text file.
- The "Parts" folder contains the actual ship-building parts you would see in the ship builder interface.
- The "Plugins" folder contains the plugin .dll files that would support the plugin system implemented by the addon.
- The "Source" folder is just the source code, and is only needed by modders wanting to see how the addon was made and maybe attempt to modify it (with permission/attribution, of course).