From the Online Etymological Dictionary:
"Eucharistic service," Old English mæsse, from Vulgar Latin *messa "eucharistic service," literally "dismissal," from Late Latin missa "dismissal," fem. past participle of mittere "to let go, send" (see mission); probably so called from the concluding words of the service, Ite, missa est, "Go, (the prayer) has been sent," or "Go, it is the dismissal."