The Most Newbie WordPress Question Ever

New to WordPress, blogging, web hosting. I see that there is a program I can download from the download page. But when I chatted with one of the web blog hosters recommended by WordPress, they said I didn’t need to download any software, that they do it all on their end, and that the blogging takes place in a web-based environment, not on my desktop computer. I had been under the impression that you write your blog on your desktop/laptop and then upload it to your site at the web host. Can anyone please clarify?

Also, I do have a registered domain name (through Register.com). Is this of any use to me while I’m just only blogging? Or do I just hold onto it until I design and build an actual web site — one that does a lot more than just blog? (E.g., a professional site with resume, work samples, etc.)

Thanks a lot.

Giraut