I set up a 90 day free joomla demo site using my "domain name with .demojoomla.com", and I also purchase a standard licence with Jomsocial-after the 90 days I sign on with SiteGround for hosting my site-my site would then only have my domain name.com-would I be able to simply move the site set up under my "domain name.demojoomla.com" along with the Jomsocial extension over to the site which now only has my domain name or would I have to pay a new 6 month licence.

HI, Stan.

User "allsta8842 " doesn't have any JomSocial license.
If you purchased it using different account - you should use that account in order to receive support.

If you installed JomSocial on one domain but after some time moved to another - just post a supported domain change request on our forum in Billing Issues category. You don't need to purchase another license in this case.