Archives are generally known as a way to look back into the past of any website. You cannot change or update the content but it’s there. It is very seo friendly because search engine bots like to surf through archives more than the site itself since everything is in order and looks great. A forum archive is basically the same thing. It’s just a link that keeps all the posts, threads, boards, and members in a cartain page of the forum for the search engine bots to enjoy themselves.
Archives of all sites are taken once in a while and saved by Archive.org so if you are wondering what site was on your domain 3 years ago, you can check it there. But if you just newly registered the domain and it was never dropped, don’t bother looking because you will most likely come up empty. If you have a vBulletin forum like I do then you have no worries since the software saves an archive and it is a built in feature. If you have other forum softwares be sure to see if there is a modification available to addon that feature because that can help you out later on with search engines.