What does it usually mean?
A stalemate due to the Fáilure of any party or bloc of parties to assemble a majority of TDs in favour of a government formation
What does it mean in this election?
The prospect that, based on all available opinion polls, the current FG/Labour government will fail to hold on to its majority, or even come close enough to a majority to form a broader coalition with other parties, or to govern with the support of independents. Add to that the fact that no alternative coalition is on offer to voters at this election, leaving the prospect that the new Dáil will not elect a government when it meets for the first time on March 10th.
Where did it come from?
The phrase is believed to have its roots in the 19th century American phrase “hung jury”, derived from the sense of “hung” to mean caught, suspended or delayed, as in “I got hung up at the office”.
Does the phrase occur much in Irish politics?
Sometimes, but it's really not relevant. It's more associated with two-party systems, such as the US and UK, where the usual outcome of an election is a clear majority for one side or the other. In such countries, the prospect of no outright winner is often viewed with horror, with words such as "chaos" and "stalemate" bandied about. With a system of proportional representation such as ours, an outright majority is a rarity (although Fianna Fáil has managed it on a few occasions).
So why is it such a big deal now?
Largely because the tectonic shifts in Irish politics since the 2008 crash have changed the landscape to such an extent that it's quite tricky to predict what will happen once the election is over. The old binary of Fianna Fáil (plus a small party or independents) versus Fine Gael plus Labour seems to be gone. Fianna Fáil and Fine Gael between them now only get the support of half the electorate. And Labour will lose most of the seats it won in 2011.
Should we be worried? Hard to tell. The received wisdom is that there will be huge pressure on the parties to come to some sort of agreement. Nobody (particularly the exhausted TDs who've just spent huge amounts of energy and money getting elected) wants another election straight away. The most likely permutations right now seem to be either a Fine Gael – Fianna Fáil coalition or a Fine Gael-led minority government relying on Fianna Fáil for support from outside on big economic issues (on the model of Fine Gael's 1987 Tallaght Strategy, when it supported Fianna Fáil). The main impediment to the former is likely to be the sincerely held view of many within Fianna Fáil that they would be signing the party's death warrant if went in as junior partner with Fine Gael and left Sinn Féin as the main party of opposition. The Tallaght Strategy option might be more attractive. Plus it's worth noting that, while a decision to go into government needs to be ratified by Fianna Fáil at a special party conference, that would not be necessary to support a minority government from outside.
What's the alternative? Hard to say. Sinn Féin and some (but not all) of the other left parties are campaigning on a broad Right 2 Change platform, but even they are not trying to argue they're a plausible alternative government this time around. There is still a chance that Labour and Fine Gael will perform a bit better than predicted, and will manage to cobble something together with some of the independents and small parties who are predicted to perform well.
Could this take a while? It could, and has done in the past – for example, following the general election of 25th November 1992, Albert Reynolds was only elected Taoiseach on January 12th, 1993, following intense negotiations on the formation of a Fianna Fáil – Labour government. In other countries, it can take a lot longer; Spain is still awaiting a government more than two month after it held its election. Belgium had to wait 589 days in 2010-2011 for a government to be formed. The general view was that everybody managed just fine without it.