Hard to throw a stone in Galway without hitting a great session. Surprised this was posted in Feb and no one's responded yet.
I was there for a few months in 2007, so I don't have all the latest details, but generally Tigh Coili's, Taaffe's (right across from each other right in the center of the pedestrian area) are among the top, though my personal favorite is The Crane on Sea Rd across the river. Last I knew, Sean Ryan comes up from Tipp to lead the Sunday afternoon session there each week. (looks like that's still the case, according to the website:
http://www.thecranebar.com/ )
If you're a beginner-to-intermediate player and are looking for a friendly, welcoming session for all levels, check out
http://www.dustybanjos.com/ Great folks, great tunes.
This is definitely not an exhaustive list. Most trad pubs will have sessions at least once or twice a week, though the ones above probably have something going on just about every night, plus some in the afternoons.
Enjoy!