How can I move my installation to a different directory?
Update the Board URL in the ACP -> Options -> Basic Board Information and move the files to the new directory. You may need to restart the server for the change to take effect.
How can I move my installation to a different host?
Backup the existing database and server files. Copy them to the new server and update the library/config.php file with the new database details. Lastly, update the Board URL in the ACP -> Options -> Basic Board Information, if it has changed.
Note that if you are moving to a different type of web server and have Friendly URLs enabled, you will need to make the appropriate changes. Refer to the Friendly URLs section of the XenForo Help Manual for more details.