"Hymn of the Virgin Mary," c. 1200, from Latin third person singular of magnificare, from magnus "great" (from PIE root *meg- "great") + combining form of facere "to make, to do" (from PIE root *dhe- "to set, put"). So called from the opening of the Virgin's hymn (Luke i.46, in Vulgate Magnificat anima mea dominum "My soul doth magnify the Lord") which is used as a canticle.
To correct Magnificat (before one has learnt Te Deum) is a 16c.-17c. expression for presumptuous fault-finding, attempting that for which one has no qualifications.