In this case, if you get a hosting package from 123 reg, then no, by uploading your site to the space there, you are hosting your site on their servers.
To be able to host a site yourself, you need to go to System Preferences and click on sharing and then click on web sharing or rather than using Mac OS 10.5 you need to purchase a copy of Mac OSX server and use this.
It is much simpler just to get a hosting package from 123 reg with your domain name and publish and upload to their server with Cyberduck.
It is a lot of work to run one a machine as a server and you need two computers as well.