Many people have created projects that control an Arduino board over the Internet with web pages, cloud services and other GUI-style interfaces. However you can still send and receive basic text with an Arduino over the Internet using an Ethernet shield and the right software - in this case Telnet (or other terminal software). At the FIUADY website they have documented and demonstrated this by showing how to control digital output pins using such as method.
And although this isn't a graphical interface, the Arduino can still respond with text sent back to the terminal, just like "in the good old days". So to get started, click here. And for more, we're on twitter and Google+, so follow us for news and product updates as well.
If you're looking to recreate this remote control method for your Arduino project, consider our Freetronics EtherTen. Apart from being fully Arduino Uno-compatible, it has onboard Ethernet, microSD socket, full USB connection (no pesky FTDI cables needed) and supports optional Power-over-Ethernet: