Celtic music is pretty big right now. It was growing even before the whole "Riverdance" thing several years back and isn't slowing down any.
Celtic music is simple folk music; it's dance music and is meant to be enjoyed in a small hall or pub. Most tunes are "session" tunes; usually a combination of three different tunes. Each tune runs a few dozen bars before segueing into the next tune.
If you want to learn some of these session tunes, look no further than Francis O'Neill. O'Neill published thousands of these tunes in several books in the early 1900's. "Music of Ireland" was the most popular -- over 1800 different dance tunes.
You can still purchase these books today -- or you can download their contents for free from several source.
Free Celtic Sheet Music is a good place to start -- you can download the contents of O'Neill's three major works: "Dance Music of Ireland", "Music of Ireland ", and the rare "Waifs and Strays of Gaelic Melodies". Combined, these three volumes contain over 3,000 celtic tunes; you can download standard notation transcriptions. Many pieces also offer accompanying MIDI files.