First, a very brief history lesson: The whole of Ireland became part of the U.K. in 1801, but after a century of British rule and an uprising (sound familiar?

), the island was partitioned in 1921, the six counties in the north remaining with the U.K. and the other 26 counties becoming a completely independent country.
The result of this is that some things in the Republic of Ireland are very similar to the U.K. while others can be quite different. Some of the differences are quite trivial and merely act as a reminder that one is not in Britain, e.g. the mixture of familiar British brands and local Irish brands in stores, the different color mailboxes, the road signs (which are a curious blend of British and American with a few local oddities thrown in for good measure), the bilingual name signs, and so on.
There are more far-reaching differences though. After many years of using the Irish "punt" or pound (which was not the same as the pound Sterling) the RoI adopted the euro currency a couple of years ago. Ireland has its own tax system, welfare system, and national health service, which although similar in principle to the British systems each have their own quirks. Depending upon your circumstances, you might be better or worse off compared to the U.K.
Real estate prices have soared in recent years across Ireland, but prices in the City of Dublin are absolutely insane (more expensive than London!). I know someone who has just moved there from his family home in Cork for a short-term work contract and he says it's costing a fortune just for a tiny apartment. Unless essential for work, I honestly can't see much to recommend Dublin anyway. There are
much nicer places in Ireland to live.
Re visas, it might be worth checking for any Irish ancestry in the family. I believe that the RoI offers easier immigration in this respect than Britain, even if the Irish link is three or four generations back.