And you do that with whoever you bought your domain name from. Often if you have hosted/shared webspace (sounds like you might do - using cpanel) then your host is probably also your DNS provider. But perhaps not?
They can be separated (DNS and hosting). Usually they'll have a page where you can change the settings yourself. You'll need to adjust the A record to point straight to your elastic IP (currently it probably points to your hosted webspace). If you are no longer using your webspace then you can ditch that bit and just leave the domain with them, or you could tranfer your domain to someone else (Amazon do that too but not sure how competative their pricing is).
If you still wish to use your current webspace and have your TKL server as a subdomain, I'm pretty sure you need to add a new A record for the FQDN of your TKL server pointing to the elastic IP. And leave an A record pointing to your existing server. So it'd go something like this:
www.my_domain.com. IN A xxx.xxx.xxx.xxx
moodle.my_domain.com. IN A yyy.yyy.yyy.yyy
Where xxx.xxx.xxx.xxx is your existing shared server IP and yyy.yyy.yyy.yyy is your TKL Moodle elastic IP.
And if you want the base domain to be the same as www then add this CName record too (possibly already exists):
my_domain.com. IN CNAME xxx.xxx.xxx.xxx
In my experience most domain providers only require you to select the record type, fill in the domain name and the IP address, but if for some reason yours needs you to do the full line, don't forget the trailing dot after the domain name.
If you don't want to do a subdomain and want it to be a subfolder instead (ie www.my_domain.com/moodle not moodle.my_domain.com) then I'm not 100% sure how you do that.
You need to adjust your DNS records
And you do that with whoever you bought your domain name from. Often if you have hosted/shared webspace (sounds like you might do - using cpanel) then your host is probably also your DNS provider. But perhaps not?
They can be separated (DNS and hosting). Usually they'll have a page where you can change the settings yourself. You'll need to adjust the A record to point straight to your elastic IP (currently it probably points to your hosted webspace). If you are no longer using your webspace then you can ditch that bit and just leave the domain with them, or you could tranfer your domain to someone else (Amazon do that too but not sure how competative their pricing is).
If you still wish to use your current webspace and have your TKL server as a subdomain, I'm pretty sure you need to add a new A record for the FQDN of your TKL server pointing to the elastic IP. And leave an A record pointing to your existing server. So it'd go something like this:
www.my_domain.com. IN A xxx.xxx.xxx.xxx
moodle.my_domain.com. IN A yyy.yyy.yyy.yyy
Where xxx.xxx.xxx.xxx is your existing shared server IP and yyy.yyy.yyy.yyy is your TKL Moodle elastic IP.
And if you want the base domain to be the same as www then add this CName record too (possibly already exists):
my_domain.com. IN CNAME xxx.xxx.xxx.xxx
In my experience most domain providers only require you to select the record type, fill in the domain name and the IP address, but if for some reason yours needs you to do the full line, don't forget the trailing dot after the domain name.
If you don't want to do a subdomain and want it to be a subfolder instead (ie www.my_domain.com/moodle not moodle.my_domain.com) then I'm not 100% sure how you do that.