Hopefully someone who understands Irish politics will explain what this all means.
Sounds like they are the socialist party.
A political wing of IRA - communists.
Sinn Fein is more or less the political wing of the now most defunct IRA. They run candidates on both sides of the border for federal, legislative and local races.
They support the idea of a united Ireland. their ideology is pretty left wing.
Wikipedia has an surprisingly good article about them:
https://en.wikipedia.org/wiki/Sinn_F%C3%A9in
Three out of every four Irish voters voted for someone else, but since they have a razor thin plurality, Sinn Féin won.
They have to forge a working coalition out of other leftist parties to have a role in government. But their role with the Provisional IRA puts off a lot of people and will make it hard for them to cobble together a government.
Conversely, the other parties may try to forge a coalition to keep them out, well see.