sounds like you are a little confused over something simple
domain names can be purchased from places like this
http://www.names.co.uk/, as well as a domain name you will need website hosting, I believe that company does that as well but they are not the only one so shop around
technically people do own the internet as web hosting companies either buy or lease servers and rent out space on those servers to those that want websites,
the domain name is an ip that has been converted via dns ( a domain name server ) into a domain name to make it simpler for us to remember ( its simpler to remember words that groups of numbers )
when you sign up with a website host, the host will either setup the domain for you or if you already own a domain it can be 'pointed ' to their server by using something like ns1.myhost.com or ns2.myhost.com ( the myhost.com part is usually your web hosts name and ns stands for nameserver )
I hope it clears it up a little for you, if you have more questions try a web design forum, I own one
http://webprocafe.com but there are quite a few out there, find one you feel comfortable in and ask questions.
p.s when you purchase your domain name they are owned by you unless you let a renewal lapse, its call intellectual property rights, no-one else can lay claim to your domain, even the web host you registered it with.