Domain names
When you enter a website url into the browser's address bar, it uses DNS to look up the IP address for the server.
DNS is the domain name system. Domain names are registered at a registrar, and are bought, leased, and sold like real estate.
Check out namecheap and hover for domain registration.
When you buy a domain and assign it to a server, it can take a while (hours to days) to propagate through the DNS systems on the internet.
Web hosting
Many sites are hosted in the cloud, on someone else's servers. This makes it easy to rent a slice of hardware and only pay for what you need. There are a variety of options now to choose from.
Your site will have software and server requirements to install and maintain. Many web hosts have the common ones installed to handle most applications. Bigger apps may need custom infrastructure.