Step 1

Sign up for hosting somewhere. You need to have PHP and MySQL installed, at least.

Dreamhost will work nicely, and quite often, you can signup, with a unique domain name for just 10 dollars, for the first year.

There are many other hosts to signup with. It will often run about 5-15+ dollars a month. If you are not getting mega ammounts of traffic, a host somewhere within this range should suffice.

Step 2

Install MediaWiki. This can have a bit of a learning curve, but once you get the hang of it, it is not super super difficult. See:
http://www.mediawiki.org/wiki/Manual:Installing_MediaWiki

Step 3

Get the XML dump of the wiki that you wish to fork.

Many can be found at:
http://download.wikimedia.org/backup-index.html

If you want to fork Commons, then see:
http://yousefourabi.com/blog/semantic-web/...ages-with-wikix

If the wiki you wish to fork does not have an XML dump available, then you still could fork the wiki, but it could be a pain. You would have to get the page list at [[Special:Allpages]] and then either export all pages manually at [[Special:Export]] (with a large wiki, this could take a long, long time) or else setup some sort script to do it (find it using Google, if it exists, or write it yourself).

Step 4

Import the dump. See:
http://www.mediawiki.org/wiki/Manual:Importing_XML_dumps

Appendix

Configure the Wiki

Then of course, configure MediaWiki at sometime in here. You may want to Import [[MediaWiki:common.css]] from the big wiki.

You can enable the use of images from Commons. See:
http://www.mediawiki.org/wiki/Manual:$...oreignFileRepos

A pretty decent overview of other good to know configurations are at:
http://wiki.dreamhost.com/MediaWiki

You may need to tweak the configuration over time. See:
http://www.mediawiki.org/wiki/MediaWiki

The MediaWiki wiki has a lot of information on it, and is organized perhaps pretty reasonably.