The Lonesome Boatman by the Fureys. Perhaps one of the best Irish songs ever.
Flogging Molly - Black Friday Rule. Flogging Molly is a blend of traditional Irish music with heavy rock/punk rock. Between the dark, emotional lyrics and the unique sound, I describe them as, "Johnny Cash meets Ireland." This song is one of their "anthems" if you will, but they have a lot of soft, somber songs. Some to check out are "If I Ever Leave This World Alive," "The Son Never Shines on Closed Doors," "Don't Let Me Die Still Wondering(which is actually a Johnny Cash tribute,) and "The Story So Far." FM is by far my top favorite band and I try to spread them to everyone I can.
Streetlight Manifesto - We Will Fall Together. SM call themselves "Hardcore Jazz," mixing heavy punk rock with a combination of jazz, European and Mexican. Most would call them "Ska" since they are punk rock using trumpets, saxes, and trombones, but they hardly have any "Ska" sounding songs. You can definitely hear a lot of Jazz and European influence in their music.
EDIT:
Here's another one.
Devil Makes Three - St. James. Opened up for Flogging Molly one year and I fell in love with them. Kind of like Mumford and Sons, only they are actually good. Good 'ol Americana/Bluegrass.