What is a vigil mass?

A vigil mass is a mass said on the day before a major feast day. Vigil masses are "waiting" masses, meaning that the approaching feast day is of such import that the Church desires the faithful to take a day to prepare themselves to celebrate it fully, with true hearts and clear dispositions. A vigil mass day is therefore usually a day of fast and/or abstinence and the mass itself has propers - specific prayers - that call the soul to purify itself in preparation of the feast tomorrow.
In the old times, Vigils were there because the Sabbath started from sundown on the (now called sixth day) and ended at sundown on the seventh day.