From the Reformed perspective, what is the meaning of the tearing of the veil in Matt. 27:51? Matt. 27:51,


  "And, behold, the veil of the temple was rent in twain from the top to the bottom; and the earth did quake, and the rocks rent;"  (KJV)


Also from Mark 15:38,


  "And the veil of the temple was rent in twain from the top to the bottom."  (KJV)


Others have maintained that the veil of the temple was different than the veil of the tabernacle; and that the veil of the temple referred only to a first veil hung before the entrance to the temple from the inner court.  

But, the Pulpit Commentary at BibleHub offers:


  " The veil of the temple (τοῦ ναοῦ). There were two principal veils in the present temple - one between the vestibule and the holy place, and one other which is that here referred to, a constituent part of the edif[i]ce. This was the veil between the holy place and the holy of holies, which was moved aside only once a year to admit the high priest to the shrine on the great Day of Atonement (Exodus 26:33). It was large and costly, some sixty feet high, and made of rich materials. "  Source: here


And, Vincent's Word Studies offers:


  "According to the Rabbis this was a handbreadth in thickness, and woven of seventy-two twisted plaits, each plait consisting of twenty-four threads. It was sixty feet long and thirty wide. Two of them were made every year, and according to the exaggerated language of the time it needed three hundred priests to manipulate it. This veil was the one which covered the entrance to the holy of Holies, and not, as has been asserted, the veil which hung before the main entrance to the sanctuary."  Same source as above.


The interlinear Greek uses "σκηνὴ" or "skene" for both "tabernacles" in Heb. c. 9 which shows no distinction in a temple vs. a tabernacle for the holy and most holy places.  But, in Matt. 27:51 the Greek is "ναοῦ" or "naou" for "temple". 

Were there two veils, and if so which one was torn?  And, what is the significance of the torn veil?
This subject is dealt with very thoroughly in The Temple and the Church's Mission: A Biblical Theology of the Dwelling Place of God (New Studies in Biblical Theology)  by Greg Beale.

From a Reformed perspective, the temple and tabernacle are interchangeable as the dwelling place of God. The curtain that was torn was the curtain into the Most Holy Place, the manifested presence of God, his throne on earth. (cf. 1 Sam. 4:4 Where God is 'enthroned between the cherubim' of the Ark.) Heb. 9:3 "Behind the second curtain was a room called the Most Holy Place". The first curtain separated the Holy Place, which was entered daily, from the courtyard. 

The significance is awesome. After the fall, Adam &amp; Eve were driven out of the original tabernacle - or divine dwelling - which was Eden. Before they were driven out, God promised a Savior who would undo the Fall (Gen. 3:15). God instructed Moses to build a dwelling place, but even then it was unapproachable except by the High Priest, one day a year. Reformed Theology has always understood the temple to be symbolic, and this symbolized that access to the presence of God was not yet available.

When Jesus came, God became flesh and made his dwelling among us, lit. 'tabernacled among us' John 1:14. When the body of Christ was broken, all separation was broken with it. We now, though vile, arrogant, and filthy, can enter the very presence of the Living God, and not just once a year but at any moment. This is exactly what Hebrews goes on later to say "Therefore, brothers and sisters, since we have confidence to enter the Most Holy Place by the blood of Jesus, by a new and living way opened for us through the curtain, that is, his body, and since we have a great priest over the house of God,let us draw near to God with a sincere heart and full assurance of faith." Heb 10:19-22