Once you have download the file, either a Zip or a RAR archive you will need a tool like WinZip, WinRAR, 7Zip, etc to unpack it in your X3 Terran Conflict game folder.

Start X3 and load your save game (or start a new game) and if you are installing a script everything should works by now. If you're installing an AL Plugin you will then need to go to your Artificial Life menu ( [Esc] / Gameplay / Artificial Life ). Check for the line relevant to the plugin you've just installed and enable it. You will usually receive a confirmation window telling you everything is ok.

Except if specified otherwise in the release notes you'll just need to unzip the new version over the previous one, and when a game is loaded the plugin will upgrade itself. Sometimes scripts using hotkeys will reset their shortcut during upgrade.