Besides the campaigns shipped with the game (see MainlineCampaigns) you can play a wide variety of user-maintained campaigns (UMC).
The Campaign Server
Most user-maintained campaigns can be obtained through the Wesnoth campaign server. The campaign server is accessed from within the game itself. To use it, select the Add-Ons entry on the main menu screen. Accept the default server campaigns.wesnoth.org to connect and download a new campaign from the official server.
Some files are large and the download might time out and get interrupted if your network is slow. Remember also that the files are usually zipped (see below) so you might need to unzipped them before being able to play them.
Command Line Campaign Client
Included with Wesnoth source package is a Python command line program utils/campaigns_client.py which can access the campaign server to install and update add-on campaigns. For updating 1.3.x or 1.4.x add-ons, use one of the following commands:
campaigns_client.py -p 1.3.x -f ~/.wesnoth/data/campaigns campaigns_client.py -p 1.4.x -f ~/.wesnoth/data/campaigns
(Use your actual Wesnoth user data directory if not ~/.wesnoth)
After modifying the command(s) for your needs you may have them run automatically by cron or Task Scheduler to periodically update add-ons.
Use campaigns_client.py --help to check its detailed usage. Note that campaigns_client.py uses Python modules that are included in the source package, but may not be installed on your system. If you run the program from the root of the source package, it will be able to use the modules without needing them to be installed.
The Wesnoth wiki no longer carries detailed information on UMC, because the campaign-designer community is now large enough to make that impractical; the information changes too rapidly and would soak up huge amounts of maintainer time to keep properly current. To find out what is available and its state of completion, we suggest you examine the list of add-ons on the Wesnoth campaign server and then read the corresponding threads in the Scenario & Campaign Development forum.
You might also want to visit Wesnoth UMC Dev, a site for collaborative UMC development with its own SVN repositories.