From Catholic perspective, what is the meaning of the tearing of the veil at the death of Christ? Matthew 27:51,


  And behold, the veil of the sanctuary was torn in two from top to bottom. The earth quaked, rocks were split (NAB)


Mark 15:38,


  The veil of the sanctuary was torn in two from top to bottom  (NAB)


What do the Catholics teach about the significance of the veil which was before the Holy of Holies in the temple being torn from top to bottom?
St. Thomas Aquinas himself thinks only the outer veil was rent; he says in his Commentary on St. Matthew's Gospel:


  And it ought to be observed that in the Temple there were two veils, namely, in the tabernacle, because there was the veil within the Holy of Holies, and there was another one, which was not in the Holy of Holies. And these two veils signify a twofold veiling, because the inside veil signifies the veiling of heavenly mysteries, which will be revealed to us: for then we shall be like to Him, when His glory shall have appeared. The other veil, which was outside, signifies the veiling of mysteries which pertain to the Church. Hence, the outer veil was rent, but the other one was not, to signify that mysteries which pertain to the Church were made known by Christ’s death; but the other veil was not rent, because heavenly secrets still remain veiled. Hence, the Apostle says: “But when Israel shall be converted to the Lord, the veil shall be taken away” (II Cor. 3, 16). Hence, by the Passion, all mysteries, which were written in the Law and the prophets, were opened, as it is stated: “Beginning at Moses and all the prophets, he expounded to them in all the scriptures the things that were concerning him” (Lk. 24, 27). Or the rending of the veil signifies the dispersion of the Jews. And because their glory was in the veil that was rent at the Lord’s Passion, it was signified that all glory was divided from them.


The Fathers seem to think only the outer veil was rent:

St. Thomas Aquinas, in his collection of the Fathers' commentaries on the Gospel, the Catena Aurea on Mt. 27, quotes these:


  St. Augustine: de Cons. Ev., iii, 19: The wording sufficiently shews that the veil was rent just when He gave up the ghost. If he had not added, "And, lo!" but had merely said, "And the veil of the temple was rent,: it would have been uncertain whether Matthew and Mark had not inserted it here out of its place as they recollected, and Luke had observed the right order, who having said, "And the sun was darkened," adds, "And the veil of the temple was rent in twain;" [Luke 23:45] or, on the contrary, Luke had returned to what they had inserted in its place. 
  
  Origen: It is understood that there were two veils; one veiling the Holy of Holies, the other, the outer part of the tabernacle or temple. In the Passion then of our Lord and Saviour, it was the outer veil which was rent from the top to the bottom, that by the rending of the veil from the beginning to the end of the world, the mysteries might be published which had been hid with good reason until the Lord's coming. "But when that which is perfect is come," [1 Cor 13:10] then the second veil also shall be taken away, that we may see the things that are hidden within, to wit, the true Ark of the Testament, and behold the Cherubim and the rest in their real nature. 
  
  St. Hilary: Or, The veil of the temple is rent, because from this time the nation was dispersed, and the honour of the veil is taken away with the guardianship of the protecting Angel.


The Catholic Haydock Commentary on Mt. 27:51 thinks both the inner and outer veils were rent (my emphasis):


  Ver. 51. The veil of the temple was rent. As there were in the temple two parts of the sanctuary, so there were two veils, or partition walls. The first sanctuary, called the holy, was separated by a veil from that part of the temple called the court of the Israelites. Into this outward sanctuary, called the holy, entered every day the priests that were in office. The second interior sanctuary, called the holy of holies, was also separated from the outward sanctuary by another veil. And into this holy of holies, no one was to enter except the high priest, and he but once a-year. Both these veils seem to have been rent at Christ's death: and by their being broken down, was signified first, that the ceremonies of the ancient law were to be abolished by the law of Christ; and also that heaven should be open to all. …