I'm not too politcally well-versed, but it seems that never stops most people from giving their opinions! So here's a cynical and ill-informed rundown:
Fianna Fáil and Fine Gael: the two main political parties, bunched together here because they're the same damned party. Really. Shortly before Ireland became Ireland, they split up because one side was in favour of a certain treaty with Britain and the other wasn't (i.e. one side insisted on fighting for the entire island, the other one said we were glad to get most of it). This was close to a hundred years ago, and pretty much ever since, one or the other has been in power, despite there being seemingly no difference between the two.
Labour: the social democrat party that faced ruin in the last election for totally bailing out on their voters (they went from 37 seats in 2011 to 7 in 2016). They're going to go the way of the Progressive Democrats - a party with a number of seats big enough to be needed by FF/FG to stay in government, but not big enough to actually do anything or live up to their campaign promises, so they drop out of favour with their voters and then disappear entirely. As in, the PDs don't exist anymore. We'll see if Labour ever spring back.
Sinn Féin: terrorists in suits, according to some. Historically associated with the IRA, and their main goal is the re-unification of Ireland. Fairly left-wing in most other regards, but often considered contrarian rather than being taken seriously - depending on who you talk to, obviously. I think I actually voted Sinn Féin in the last election (we use proportional representation, so I mean I gave them 2nd or 3rd preference, I can't remember.)
Greens: Who? Oh the lads who want to put windmills in
We also have a smattering of Independent TDs (=Teachta Dála = MP, basically), and of course the left-wing Solidarity-PBP - formerly the AAA-PBP; the Anti-Austerity Alliance and the People Before Profit joined up, but it doesn't exactly roll off the tongue.
And there was Renua last year, a new political party that more or less consisted of a few politicans having a conference and making vague statements about what they'd do, then getting no seats.
I don't really know enough about politics to give an actual opinion, though, which makes conversation at election time rather awkward.