ZOMG is an extra-internety way of typing OMG -- or for those of you with a life outside the internet, Oh My God!
The initial 'Z' is a faux-typo, arising from a poor typist hitting the z key instead of the shift key. Because of this, it is often written zOMG, although this has become less common as the meme spreads. ZOMG was originally used as a sarcastic or mocking exclamation, but it is increasingly being treated as a self-mocking and/or silly form of OMG, rather than openly sarcastic. It has since been accepted into the internet-slang-from-typos phenomenon, along with teh, pwned, and the ever-popular !!!!!111oneoneoen.
The addition of an initial 'Z' is also common in ZOMFG, although it does not seem to be spreading to other initialisms, such as LOL and WTF.