Microsoft Exchange is a mail server, with some other (nice) features.
OS X Server includes the Postfix mail server, with some other features.
"Cloud technology" is whatever the marketeer you're chatting with wants it to mean. What it usually means is some variation of "remote servers I don't own and don't control, but still have to pay for". In an earlier and less-cloudy era of marketing, this was known as "hosted services" or a "service provider" or other such.
Your email services are not based on "cloud technology" nor on the unicorns and rainbows that the marketeers are fond of including on the slideware, it uses a domain name and a mail server.
If you want to move a domain name that you have registered over to a mail server that you control, or to a hosted mail provider (aka "cloud"), that's easy. That's an update to the DNS records and some set-up work (by yourself or the hosting provider), and you're good to go.
If you're thinking of moving to Apple iCloud, then I don't know if Apple allows you to associate your own domain name; you're using the @me.com domain. AFAIK, Apple does not support this. Other hosting providers do allow this, however.
If you want to move a domain name you don't have registered, well, that's not going to happen. You can often set up mail to forward to another provider, but (if it's not your domain name), you don't get to decide where it's hosted.