Joe has documented the process of monitoring an Arduino digital input pin over the Internet as part of a garage door system. Although he hasn't moved forward with the entire project, his article is excellent as it descibes the complete software and hardware process for sending information from an Arduino board over the Internet.
In his example the status of a digital input pin is returned. You can then build upon Joe's code and make your own monitoring solutions. An ideal base for these projects is to use our Freetronics EtherTen or EtherMega boards - the only Arduino Uno/Mega compatibles with onboard Ethernet, microSD card socket and optional PoE support:
For more information and to get started with your own Internet-connected project, visit Joe's site here.