Who should own the Internet? (ICANN? IANA? ITU? countries?)
ICANN is a non-profit organization that works to ensure the stability of the Internet's namespaces.
IANA is a department of ICANN that maintains the IP address space. Groups of IPs are delegated to Regional Internet Registries (RIRs), which in turn delegate groups of addresses to Internet Service Providers (ISPs).
Domain Name System
Originally domain names were stored in a file called HOSTS.TXT which mapped hostnames to numerical addresses. Hosting, distributing and updating this file was a burden, so it was replaced by the Domain Name System. Under DNS, registered companies compete in providing name-to-address translation using name servers. Domain name registrars are accredited by ICANN and are obligated by ICANN to provide WHOIS support.
Top-level domains (TLD's) include .COM and .NET and are the last part of a domain name. Top-level domains are governed either by single registrars (e.g. .DE) or groups of registrars (e.g. .COM). Generally, country code TLD's (ccTLD's) are operated by private companies or government authorities within their corresponding country.