Q. What does it mean when it says "the veil of the Temple was torn?"

A. This is a very mystical expression and let me explain some things. The expression is found in many places but I assume you are referring to the phrase in Mt 27.51. The veil that was torn was the outer veil leading into the Holy Place of the sanctuary, not the Holy of Holies. That had two veils and it could be entered by walking between the veils. This outer veil of the gate on the Ulam (porch) of the sanctuary was made of four colors. Blue indicating the heavens; linen, or flax, was brownish in color symbolizing the earth; scarlet indicating fire and purple stood for the sea. This veil had stars and the known planets on them and it was called the "panorama of the heavens." This gate to the sanctuary where the veil stood opened by itself for 40 years after the death of Yeshua. This curtain, or "the heavens", was opened on the 9th of Av and other occasions for the people to look into the holy of holies and so the veil being taken away was done before (Josephus, Wars, Bk 5; Hertz Siddur p. 959; Kaplan,Torah Anthology, p 91). The "tearing" or opening of this veil meant that the deeper things of God will now be understood, or open. When Stephen in Acts 7 or John in Rev 4 saw the heavens "opened" it meant access to the hidden, the mystical. It meant you are about to understand the deeper things of God and in Stephens case he was going there in person. So, with that in mind when it says in Psa 84.10 that "I would rather stand at the threshold (door) of the house of my God" now you know why. To be a "door man" in the house of God means that you can "open" up to the people the mystical, hidden, deep things of God. So, when Yeshua died on the cross and this veil was torn, opened, so the people already knew what this meant and that God was revealing something very deep and access to the things of God to be revealed was now accomplished for anyone who believes. There were many, mystical things going on when this happened and the writings of the Gospels and Epistles brought out so much more about redemption, salvation, propitiation and everything associated with the blood of Messiah and what it accomplished. God gave the people a sign just at the right moment to say that He was about to take the people deeper into the things of heaven and that its mysteries were about to opened to the people, and that's exactly what happened. And not everything has been revealed yet so that door is still open to us today.I hope this answers your question.