I like the Irish Traditional Music Archive (ITMA) a lot. The whole site is fascinating to browse, but in particular the scores and notated collections are great. The scores are interactive, in that it will play the tune for you (albeit in generic piano tones) while highlighting the notes on the score as they are played, and you can slow down the playback (lower the BPM) if needed. Tunes from a few of the sources that have been mentioned above are included, and there are quite a few sources that I've never heard of, but love to explore.
Ryan's Mammoth Collection (1050 Reels and Jigs, Hornpipes, Clogs, Walk-arounds, Essences, Strathspeys, Highland Flings and Contra Dances, with Figures) is another great tune book that I don't think has been mentioned yet. It's from the 1880s, and as implied in the ridiculously long subtitle, not all of the tunes are Irish. Unlike many of the most cherished tune books though, this one is still widely available (Mel Bay has a facsimile edition that is relatively cheap). Many of the mammoth tunes are available as ITMA scores.
Gary's list of sources is awesome, and endgrainguy mentioned irishtune.info, a site that I use daily as well. Here is the bibliography, listing sources that Alan Ng indexes on irishtune.info (many of them have already been mentioned by Gary above, but many others here are also worth a look). Here is a subset of the bibliography that Ng highlights as the "Best Session Tunebooks." Ng prefaces this bibliography with some words about the benefits of learning by ear rather than books. I usually need both to learn. One of the great things about Foinn Seisiun books (mentioned by Gary) is that you can get CDs with every tune in the books. There are now 4 volumes, and while I can't find the 4th volume on Comhaltas website, if you email or call them they do have it, and they can tell you about discounts for buying sets of 3 or 4 volumes with the CDs. I guess I should mention, I don't work for them, or Mel Bay, or Alan Ng, or ITMA!
I don't disagree with anything that's been said about The Session so far in this thread, but I have to admit that I often go there first for convenience. I tend to compare with other sources later if I'm 'serious' about learning a tune, or if it is just not sounding right.
My apologies for rambling on, and for resurrecting an oldish thread (not sure if that's bad form or not, but I am a newbie after all).