It's easier that you think to control an Arduino remotely via the Internet, and the following project by Instructables user "fpoels" is another excample of this. In his project he uses an Ethernet-enabled Arduino board as the target device, which runs a sketch which generates a simple HTML control page for the digital output - which then naturally can be accessed and used to control a digital output. It's a great example and one that you can work with for your own remote-control devices. A quick demonstration follows in the video below:
For complete instructions, visit the project page. And for more, we're on twitter and Google+, so follow us for news and product updates as well.
If you're looking for an Ethernet-enabled Arduino-compatible board, consider one of Ethernet-enabled boards, such as the Freetronics EtherTen - the Arduino Uno-compatible with onboard Ethernet, microSD socket and optional PoE:
... or for more program space, I/O ports consider the Freetronics EtherMega - the Arduino Mega2560-compatible with onboard Ethernet, microSD socket and optional PoE: