Lynda has already pointed you in the direction of irishgenealogy.ie, which is a very good site indeed (and free), but be warned, Irish civil records only go back to 1845 for marriages and 1864 for births and deaths.
The site has some earlier parish records, but not many. For more, you could try http://www.rootsireland.ie/ . It's a pay site, and coverage is patchy (lots of old parish records are missing thanks to the 1922 Four Courts fire), but it's the most useful site I've found for BMDs from before the Civil Records start.
For the six counties of Northern Ireland, GRONI (the General Register Office of Northern Ireland) has access to some BMDs that IrishGenealogy doesn't have, although you have to pay for them.
As for census, the 1901 and 1911 Irish censuses are online, free in their entirety, at http://www.census.nationalarchives.ie/. They also have some fragments of earlier censuses, which are otherwise lost.
Some other very useful free sites if you're researching Belfast/northern ancestors are:
One last thing while I remember: Irish Townlands
is a very good site for understanding of Irish geographical divisions - counties, baronies, civil parishes and townlands.
Hope these are some help.