If you want to access the computer that the Arduino is attached to from anywhere, using a web interface, you will need to be running a web server on that computer. Either IIS (yuck) or Apache (great) will do. The scripting language you choose to use, ASP.net (yuck) or PHP (great), will need to be able to send serial data to the Arduino, and, possibly, get data back.
Both ASP.net (using C# or C++) and PHP can do that. Apache and PHP are free, by the way. The Micro$oft stuff isn't.