When and why did the Irish stop being a heavily discriminated against group in America?

It's important to note that not all Irish-Americans were discriminated against in the 19th century. The Protestant Scots-Irish had settled earlier and were well established in the New World by the mid-nineteenth century. They had already produced their own President, Andrew Jackson, in 1829. The Catholic Irish, on the other hand, arrived in huge numbers in the 1840's and suffered a wave of discrimination over the following two decades or so (I posted about this before, if anyone's interested).

What was the turning point for the Catholic Irish immigrants? I'd argue it was the Civil War, or more accurately the Reconstruction era. Irish labour became quite valuable in the economic boom that followed the Civil War. This had a major knock-on effect on the immigrant population, allowing them organize into unions, engage in politics in an effective way and - perhaps most importantly - send their children and grand-children to school. There was a huge expansion of the Catholic Parochial school system in the Reconstruction Era, with the result that by 1900 only 15% of Irish males continued to work as unskilled labourers. Education and organization ultimately reaped political dividends, with P.J. Kennedy (mentioned by the poster below) a good example of a successful Irish machine politician.

It was ultimately an economic revolution that helped the Irish-Americans, not a social one. These immigrants and their children found themselves well positioned during an economic boom-time, which in turn allowed them to access educational and political arenas that were previously closed to them. In effect, the Irish changed from an unwanted (and dangerous) economic burden on the established American settlers, to a valuable labour force - and with that the majority of anti-Irish discrimination ended.