The process is known as registry, and it's a part of the Immigration and Nationality Act. Those who have lived here since the start of 1972 would be eligible for permanent resident status if they can show good moral character, and would otherwise be eligible for U.S. citizenship except for the legal residence requirement.
Registry has been part of U.S. immigration law since 1929, and at that time only applied to people who had arrived here prior to 1921. It was created to help those whose entry records had been lost.
Obviously, it's enormously difficult to avail of registry as it currently stands, with its requirement of 35 years of undocumented status.
As you will read elsewhere in this issue, the Irish Lobby for Immigration Reform is hardly ready to throw in the towel in its quest to legalize the Irish, even though the comprehensive immigration reform debate has ended in Congress. And the Irish government has also expressed its desire to see the issue addressed in a proactive way.
Though dejection and disappointment is entirely understandable, perhaps Plan B will work out just as well in the end. Make sure to keep updated at www.irishlobbyusa.org.