(Note: I have tried to keep this account as factual as possible and my own views out of it, so any political slant and/or bias people read into it is as much a product of their minds as my words!)
The original IRA was formed out of the IRB (Irish Republican Brotherhood) and the Irish Volunteers during the Irish War of Independence (1918-21). It split in 1922 over the settlement with the British which divided Ireland and set up a Home Rule government in Dublin. In the Civil War that followed those who opposed the Anglo-Irish Treaty retained the name IRA.
The organisation went underground and carried out sporadic attacks throughout the next forty or so years, notably the ‘Border Campaign’ of 1956-62.
The IRA split at the start of the Troubles into ‘Official’ and ‘Provisional’ wings (1969). The Official IRA called a ceasefire after a few years (1972) and from that date on was effectively defunct as a paramilitary organisation (though they only decommissioned their weapons in 2010). The Provisionals fought on until the ceasefires of 1994 and 1996 led to peace talks in Northern Ireland and the Belfast Agreement, which set up a power-sharing regional administration in Northern Ireland. This agreement led eventually to the PIRA putting its weaponry “beyond use” and committing itself to the peaceful route advocated by its political representatives Sinn Féin.
Some of those dissatisfied with this course of action had left to form other groupings: the Continuity IRA and the Real IRA. The latter was responsible for the Omagh bombing in 1998, which killed 29 people: the deadliest single attack in the Troubles.
Just recently, elements of these and other hardline Irish Republican groups have come together to reclaim the title “IRA”. The reason for this is that name remains extremely potent, as it asserts a direct line of inheritance from the 1918-21 organization. At each split, those who have continued on have labelled their opponents as sell-outs and traitors. Republican ideology views political authority as deriving from the government of the republic declared in 1918. Defeat in the Civil War did not alter the IRA’s view of the rightness of their position: they merely acknowledged in 1922 that they could not fight on militarily. Those remaining TDs (members of the Irish parliament) who still opposed the Treaty passed their authority on to the IRA’s Army Council in 1938. Since then each IRA has claimed to be not just the true army of Ireland but also its legitimate government. (This explains why one of the cornerstones of republican ideology has been the policy of abstentionism, that is to say, refusal to participate in any other form of government, be it the Dáil, Stormont or Westminster. Over the past century, several splits can be traced back to proposed changes to this policy.)
(I originally wrote this as an answer on quora.com. The version above incorporates comments and supplementary information from Paul Devlin. The originals can be found here: http://www.quora.com/The-Troubles-Northern-Ireland/How-do-did-the-various-branches-of-the-IRA-differ)The various IRAs by Bruce Gaston