Folk Music: folkmusic.org (great resource; the successor to FolkBook).
Comhaltas Ceoltoiri Eireann, the main body supporting Irish traditional music have sites in the US (excellent) and two sites in Ireland ( atlanticisland.ie and mayo-ireland.ie). You can also see the rules of their Fleadh Ceol competition.
The Irish Traditional Music Archive.
MusicWeb Ireland is a newish Irish site with information on sessions, instrument makers, festivals, and other resources.
Dan Beimborn has a celtic music site with reviews, tunes and artist profiles.
The Dalriada celtic heritage society in Scotland is has information about instruments, celtic customs etc.
The soc.culture.scottish FAQ is a good staring point on Scottish culture.
Some examples of the wonderful complexity of celtic art (50kb file).
For Irish Resources in America, try the IrishNet directory a searchable database listings hundreds of Irish organisations and businesses in the US.
You can find loads of goodies by tunnelling around in Pat Murphy's maze of twisty little passages, which has interesting information on celtic music and various Irishy things, including a whole section on Irish and Celtic information.
For books from and about Ireland, check out Kenny's bookstore in Galway.
For information on Irish and other celtic languanges try the Gaelic Homepage.