The Arduino and Ethernet shield are set up as a web server. The web page that the server hosts allows two LEDs to be controlled by clicking checkboxes on the web page.
This video shows the Arduino and web page in action:
The LEDs are interfaced to pin 2 and pin 3 of the Arduino.
The Arduino sketch for this solution:
This code is a hack from some previous code that I wrote and so does not look exactly like the code from the sketch that the question is related to.