You can connect to internet resources with an IP address. How do you think your computer actually does it? Try this:
http://216.239.37.99/
When you do anything online you typically reference online resources (web servers, mail servers, etc) with a name. This name is technically meaningless as far as the network infrastructure of the internet goes. Something else is required.
Your computer is actually performing a couple operations when you make a request for an online resource. First it takes the name of the resource you are requesting and looks it up in the internet name "telephone book", or what is known as DNS (domain name service). This retrieves the IP address of the resource name. The computer actually makes the request using the IP address.
Now, there are some situations where a request to a single IP address will NOT yield the result you expect. One easy example is when you deal with Microsoft Internet Information Services (IIS). You can create several web sites within IIS. Each web site can have a specified name such as "mywebsite.com". You have to tell the site what this name is. When an HTTP request is made, the browser typically sends something called a "host header" in the request. It is sending the request to an IP address, and including this host header in the request. IIS receives the request, reads the host header and knows which web site to direct the request to. This concept applies to many different web server technologies. This is the technique that allows for complex web hosting companies to put many different web sites on a single server or at least on a single IP address. Otherwise you'd need a separate server for each one.
You mentioned playing quake and the example of an IP address you used. The part after the : is the port number of a request. The use of a port allows you to use several different internet protocols over a single IP. When you surf the web, the browser uses (by default) port 80. This is the standard HTTP port. There are standard ports for many standard internet protocols: 80 - HTTP (world wide web); 25/110 - SMTP/POP3 (mail); 20/21 - FTP; 23 - Telnet; 22 - SSH (remote logon); the list goes on and on.
Hope this clears up a little of your confusion.