Sorry, missed this.
Forum are forum pages- the normal ones you are looking at right now.
The archive refers to the vBulletin archive feature- most new forums keep the archive disabled these days as it serves no purpose. But 10+ years ago when search engines had difficulty indexing forums the archive was much easier for it to index. You can see an example of an archive by clicking the archive link in the footer of any page on vbulletin.org, just to the left of the "Top" link.
(Link URL copied here:
https://vborg.vbsupport.ru/archive/index.php )
Archive pages are much simpler than normal forum pages and add-ons do not usually affect the archive. Any changes you make to footer, header, or any other templates won't affect the archive either. For these reasons most people forget to put Google Analytics into the Archive code.
These days however I would recommend just disabling your archive in vBulletin settings.
As for header v. footer... In the header you *sightly* slow down the load of the page but register more visitors because no one can really stop the page loading before the header loads. In the footer there is no page slowdown but some visitors who click links really fast or click stop on their browser might not get counted because the javascipt loads last.
It really makes no difference 99% of the time.