Let us start with you connecting with the internet. For this, you require an account with an Internet Service Provider (ISP). The ISP has a range of IP addresses allocated to it by an internationally accepted organization called Network Information Centre (NIC). When you first create an account, you are given an IP address of your own by the ISP. The IP allocated to you is unique and no other computer on the internet shares your IP.
To access the Internet, you type in a URL of a resource on the Internet. The URLs are actually aliases of the IP address of the server which hosts this website. The request for the connection to this website is first processed by the ISP which maintains an updated record of look up for domain name (which is a part of the URL you type in) versus corresponding IP address. Though, most ISPs nowadays have records of all the websites, in case the domain name is not found by the look up available with ISP, it sends a request to a network called Internet backbone. Internet backbone is administered by the NIC and definitely maintains a record of all the websites hosted on the Internet. Your ISP, after retrieving the corresponding IP address, either from its own look-up or from the Internet backbone then sends the address to your browser. The browser then, looking at the URL requests the web-server for the webpage requested at the address provided in the URL. The server responds by sending packets of data stored on the webpage which is then interpreted by the browser to display the page.