You need a few things. Generally a software package to run the forum, a server to host it, possibly a database system if the forum software requires one, a domain name so people can find you on the web. This would get you most of the way there.
There are services available online for those who wish to run a system like this but don't know how to put all the pieces together. There are free options and paid options depending on your requirements. Free options will often be supported by advertisements. The previous versions of this site's message board were of this type.
If you prefer to have more control over the forum and have the ability to modify the platform to better suit your needs, you likely want a managed server environment. This is how the current forum here runs. Ngamer pays $10 each month to a company that runs a server that hosts this site and a multitude of other sites as well. He can configure his site to some degree, but the server functions are mostly out of his control.
There are dozens of forum software packages available at price ranges from free to the thousands of dollars. Choosing one depends on your needs and capabilities. Maintenance is typically required on a somewhat regular basis to keep up with security fixes, user base requests, account management, and other issues that pop up from time to time. If I had to guess, this has come out to something near 50 hours per year for me since I took over this role here a couple years ago with the move away from the Yuku board hosting. I try to keep abreast of the current situation and listen to what features users are after but stay out of the day-to-day issues that come up like deleting posts and banning people. I've mostly consulted with the community at large to determine what features to implement here, but since I was given this role by Ngamer, I pass things by him before putting them in place.
To make a forum look exactly like this one, you should install SMF 1.1.x and use the Helios Multi theme, this information is available in the bottom bar of any page here. The theme has been customized to some degree based on Ngamer's preferences. These customizations would have to be reverse-engineered by looking at the HTML output of a page and comparing it to the HTML output under the fresh installation. Note that SMF 1.1.x is now at end of life and should be updated to the SMF 2.x line. The Helios Multi theme is not available for SMF 2.x. The feeds displayed atop various pages on this forum are generated from a few sources. PR feeds come from the RSS available at the main rankings page. Twitch feeds use the Twitch API to check for streams online from our list of accounts.
I'd also like to share some stats from the forum moderation log just to paint a picture of who does what around here. This is covering the period from January 2013 to today. It looks like the people assigned to moderate the GE and PD sections are quite active, and Ngamer and come are keeping up on things as well.
COUNT(smf_log_actions.ID_MEMBER) Descending membername
282 Ngamer64
113 Thiradell
97 OversizedJimbo
93 DavidK5
81 comeasur33
66 TheFlash
35 wishiwasfamous
33 karljobst
23 Carathorn
11 AckeZ1989
11 Stefan-Persson
9 RWhiteGoose
1 Wodahs-Reklaw
1 wheatrich
1 NathanStinson
1 ExpertGamer64
One last note, this idea was also brought up about 1 year ago in a GE board thread by Malaysian Airlines Flight 370. I advise you to check out the discussion in that thread before pushing forward with any project.
If you need any help getting something set up, let me know! Or, if you have specific concerns about this site that are related to the work I do, feel free to bring them up at any time and I'll share my thoughts. Oh, and if anyone else wants to help out in an administrative/technical capacity with the forums (or rankings or main page), I'd be a good person to talk to first. I am familiar with the operation of all of these pieces and could advise you how to make sure your talents have the greatest impact.