For someone who had an ancestor living in Ireland in the first decades of the twentieth century, the National Archives of Ireland census website, www.census.nationalarchives.ie, is the obvious starting point. It is completely free, leaving plenty of scope for trial and error; every item on every return from 1901 and 1911 is searchable; and it includes images of all the returns, which are printed and follow a thoroughly consistent format. The sheer ease of use of the site means that even someone whose ancestors left Ireland long before 1901 can glean extremely useful information. Perhaps your great-great-grandfather emigrated in 1850, but if you can pick out his nephews and nieces in 1901 you're well on the way to identifying living relatives in Ireland. For more, see here.
The next step in most cases will be to General Register Office records; the State registered all births, marriages and deaths from 1864, with non-Catholic marriages registered from 1845. The Irish government site www.IrishGenealogy.ie has indexes and record images of almost all of the historic records and the Mormon website www.FamilySearch.org has a complete transcript of the central indexes to registrations up to 1922 for the entire island of Ireland, and to 1958 for the Republic. Another website, the subscription-only www.rootsireland.ie, has full transcripts of local civil registrations up to 1900, but not for all areas. See here for more detail.
Once research goes back past the start of civil registration only two classes of record are invariably useful: the property tax records of Griffith's Valuation and church baptismal, marriage and burial registers. Griffith's is freely searchable at www.askaboutireland.ie, as well as numerous subscription sites, and can provide valuable information about a family's location and economic circumstances, as well as a key to the records used in drawing it up and in keeping it up to date, many now also online. See here for more detail.
Church records, probably the most valuable source for genealogical research, are transcribed online at www.IrishGenealogy.ie , www.rootsireland.ie, www.ancestry.com and www,FindMyPast.ie. The first is free, the latter three paying. Images of the National Library of Ireland Catholic microfilms are free at registers.nli.ie. These are the images used for the transcriptions on ancestry and FindMyPast. See here for more detail.
The internet has been a wonderful boon to Irish genealogy, but it increases rather than decreases the need for scepticism. The sheer ease it brings to research can be all too seductive, masking gaps in the originals and flaws in the transcripts. Anyone tracing their ancestors online has to keep in mind that absolutely everything they see is merely a copy of the original, and has an inevitable layer of error. This means that knowing precisely what records you are looking at becomes more, not less, important. Saying 'I found it on the internet' is the equivalent of saying 'I don't know where I found it'. And if you don't know where you found information, you don't know what it means.