HELP - How Do I... discussion

I hope you will find the answers you're looking for here. What is Hosting? Is the word Hosting used to describe your Domain Name and / or your website?
Who knows, maybe someone can clear those murky waters?

Do I need to pay seperately for my domain name and website to be hosted?
It appears that where I bought my domain name (Fasthosts) they allow you to Transfer the register to elsewhere, but not the hosting. I found free hosting (Heartland) but they want you to transfer the domain name to them. Sob, sob :(