Is this how they call the IRA in Britain? Republican?
No, this is how the IRA (among others) call themselves.
Broadly speaking there are two catholic groups in Northern Ireland.
There are the “Nationalists” who believe in a united Ireland through peaceful means. Their party is the SDLP (Social Democratic and Labour Party)
There are the “Republicans” who until recently advocated murdering innocents to gain a united Ireland. This includes the NOW democratic Sinn Fein and the Real and Continuity IRA and INLA who have not renounced murder.
Can I also add that as much as 30-40% of catholics (even if they vote for united Ireland parties), have increasingly said they are happy to remain in the UK.
http://www.ark.ac.uk/nilt/2007/Political_Attitudes/NIRELND2.html
IRA = Irish Republican Army.
Republican here means believing in and upholding the Republic of Ireland. It has no connection to the Republican party in the US.