Connecting Arduino to a network need a bit of hardware - a shield in Arduino terms. http://www.nuelectronics.com/ makes some nice hardware for Arduino, including an ethernet shield. Unfortunately software and libraries that does come with their ethernet shield are weak.
Internet and open source comes to rescue again. Much more useful library can be found at http://blog.thiseldo.co.uk/?cat=14. works much, much better.
Only one big issue - it does not now nothing about MTU. If you want to send data that is longer than MTU (usually 1500 bytes) then you are out of luck.