Why do I have to pay for a URL?

I mean, who owns that? Why can’t I just have the content on my computer and host it from there? And who gets to give out and reserve URLS? I think of the internet like a giant computer, so I get that there can only be one website for each domain, like there can only be one file called Documents/Stuff.txt on my computer, but who privatizes that? I need to make a website for a school project, and suddenly am miffed by the idea of paying some faceless millionaire to host content when I could go to Fry’s and buy a TB hard drive for $80. So could some one explain,

A. How Domain registration works, and who it is that’s behind it all, or

B. How I could host my own content online. I know what I’m paying monthly for is server usage, so why can’t I just provide my own storage?

Thanks, just trying to get to the bottom of this.