The act of taking Monday off from work/class because you got crunk
so hard over the weekend that you couldn't recover in time. The joke is that you're celebrating "St Monday's Day" and so you can't come to work.
The term actually dates back to pre-revolutionary America. Benjamin Franklin makes fun of his co-workers for taking a large number of "Saint Mondays" in "The Autobiography of Benjamin Franklin".
"This, and my being esteem'd a pretty good riggite, that is, a jocular verbal satirist, supported my consequence in the society. My constant attendance (I never making a St. Monday) recommended me to the master; and my uncommon quickness at composing occasioned my being put upon all work of dispatch, which was generally better paid. So I went on now very agreeably."
-The Autobiography of Benjamin Franklin