In most common usage, the strain of Irish nationalism which endorses the use of force in pursuit of the 32-County Republic declared by the First Dail and "betrayed" by the Free State Government when it soundly beat the "Old IRA" in the Civil War. By extension, the views which lent support to the Armed Struggle during the Troubles in Northern Ireland from 1969 onwards.
Fianna Fail still declares itself to be a Republican party, which is obvious nonsense, but at least they're true to their past, unlike Fine Gael, which appears to want to draw something of a veil over the exploits of Eoin O Duffaigh and his Blueshirts pre-War.
In US terms, one of two parties which dominate the American political scene.
Republicans believe in a small federal government, freedom, no right of abortion, unlimited guns for all including assault weapons, free market policies and bombing the crap out of anyone that looks at them a bit funny.
Symbol is an elephant and they refer to themselves as the Grand Old Party. George W. Bush is a Republican which sums things up pretty neatly.