Everyone, and no one, and the ISPs.
The Internet is not a single entity. It is the collection of every single computer connected to the network. Your computer is part of the Internet too. You theoretically could host a website on it, although you don't have a URL so it would be a bit more difficult for people to find you. What happens is that when you type in a website, your computer goes and asks a special server called a Domain Name Server what the Internet address (IP address) of the computer you need to talk to is. Then, your computer sends a message to that computer.
There are a bunch of "routers" along the wires that will direct your message in the appropriate direction to get to it's destination. These lines and these routers are owned by the various ISPs. When you pay for Internet, you're paying for the ability to send your messages over these wires. ISPs have to pay for Internet too- when your traffic has to go to another ISP, your ISP has to pay them to use their wires and routers.
Since the Internet is just the connection of everybody's stuff, there's no one that really "owns" it. There are groups responsible for it, such as the Internet Corporation for Assigned Names and Numbers that is responsible for registering website names. And there are groups like the Internet Engineering Task Force and the World Wide Web Consortium who publish documents on how the Internet works so that devices, different operating systems, and browsers can all access the same content, but that's as close as you get to someone running the Internet.