Moderator: OpenTTD Developers
Click on the category for the type of add-on you want, they are listed at the very top of the page. Under the "Downloads" column where it list "<xxx> times", if you click that link it will download the newest version of the file. The files are packaged in .gz file format, unpack the file (leave it in the .tar file archive format), and then place the file in the appropriate OpenTTD directory (ai/, newgrf/, etc.). The next time you load OpenTTD the files will be visible.
There may be other NewGRFs or AIs which are not listed in Online Content (but the most popular and up-to-date files usually are), many of them are listed here on TT-Forums, others are spread across the Internet. Simply download them and unpack them, put them in the appropriate folders.
Base Music Sets: OpenMSX | Scott Joplin Anthology | Traditional Winter Holiday Music | Modern Motion Music
Other Projects: 2CC Trams | Modern Waypoints | Sprite Sandbox & NewGRF Releases | Ideabox | Town Names | Isle of Sodor Scenario | Random Sprite Repository
Misc Topics: My Screenshots | Forgotten NewGRFs | Unfinished Graphics Sets | Stats Shack | RoadTypes?