Can someone please explain the process to me, flying out from Shannon to New York on 30th December and returning on January the 4th. Do I need to sent passports off to the US embassy or what, thanks.
If you are traveling as a tourist for less than 90 days you generally do not need to apply for a Visa.
You may be eligible to enter the United States under the Visa Waiver programme. You can check if you are eligible on http://travel.state.gov/visa/temp/without/without_1990.html
Depending on your citizenship - you may or may not need a visa. Irish citizens who hold an Irish passport, as TheShark said, are generally eligible for a Visa Waiver.