The Internet is a network of networks. Nobody controls it.
Public IP Addresses are assigned by regional internet registries. ARIN is the non-profit company that assigns public IP addresses in the U.S., Canada, and Caribbean.
If you are a consumer, your IP address is assigned by your internet provider (a cable or phone company) that maps it to a public address that it got from their regional internet registry.
Domain names (which map a friendly name to an IP Address) are purchased through thousands of Domain Name Providers.
Domain names are then registered with ICANN, which manages the global Domain Name System.
The Internet's networking infrastructure is run by hundreds of private companies around the world, including cable and phone companies (which hook up end points) up to backbone networks.
Insofar as Internet content goes, there are hundreds of millions of sites owned by people like A K A Stone. Some are large. Some are small.
