“It is impossible to determine when the Roman Ritual first prescribed the use of the humeral veil. It was probably towards the close of the Middle Ages. The custom is first alluded to in “Ordo Rom. XV” (c. lxxvii). In many places outside of Rome the humeral veil was not adopted for the aforesaid functions until very recent times. It was prescribed in Milan, by St. Charles Borromeo, for processions of the Blessed Sacrament and for carrying Holy Viaticum to the sick. Its use at high Mass dates back as far at least as the eighth century, for it was mentioned, under the name of sindon, in the oldest Roman Ordo. It undoubtedly goes back to a more remote antiquity. It was slow in finding its way into use outside of Rome, and was not adopted in certain countries (France, Germany) until the nineteenth century.”–Catholic Encyclopedia
It should not be confused with the vimpa, which is the veil worn by those servers who act as crosier and mitre-bearers. Here Archbishop Burke is accompanied by three servers, two of whom are wearing vimpae.
(You have no idea how hard it was to find a picture of a vimpa!)