And Jesus again crying with a loud voice, yielded up the ghost.
 And behold the veil of the temple was rent in two from the top even to the bottom, and the earth quaked, and the rocks were rent.
(Mt 27: 50-51)
 And Jesus having cried out with a loud voice, gave up the ghost.
 And the veil of the temple was rent in two, from the top to the bottom.
(Mk 15: 37-38)
This passage marks the moment that Judaism was once and for all superseded by Christianity. In this time, the Temple contained the Holy of Holies, which is where the Arc of the Covenant would traditionally have been housed. The second Temple no longer contained the actual Arc, which had been lost when the first Temple was sacked by Nebuchadnezzar in 587 BC.
We recall Jesus saying earlier in the Gospels that the Temple would be destroyed and that He would raise it again in three days. We recognize Jesus to be referring to Himself as the Temple. This creates an interesting parallel then, because while the death of Christ is the theological destruction of the Temple, the physical temple was also 'destroyed' in the sense that the Holy of Holies was rent open.
According to the Jewish understanding, God was present in the Temple inside the Holy of Holies, so by the tearing of the veil, God shows to the Jews that He is no longer there.
God is no longer to be worshiped in the Jewish Temple. Instead the Temple is now Jesus Christ, and today He is not in the Holy of Holies, but in every Tabernacle across the world. The practice of veiling the Holy of Holies continued today, and we veil both the Tabernacle and the chalice during Mass.
The concept of veiling that which is holy has expanded somewhat from that time too, and both chapel veils for women, and the humeral veil are powerful examples of this same concept. My favourite example is that of the maniple, which is the small band of fabric which hangs off the left arm of a priest celebrating the Latin rite of the Mass. It is no longer required in the Ordinary Form, and therefore is rarely if ever seen outside of the Extraordinary Form, much to my sadness.
We also identify Jesus as the perfect Sacrifice, which was prefigured by the sacrifice of animals at the Temple. As Pope Benedict writes in Jesus of Nazareth: Holy Week:
"[...] the soldiers see that Jesus is already dead. So they do not break his legs. Instead, one of them pierces Jesus' right side -his heart- and "at once there came out blood and water" (Jn 19:34). It is the hour when the pascal lambs are being slaughtered. It was laid down that no bone of these lambs was to be broken (cf. Ex 12:46). Jesus appears here as the true Pascal Lamb, pure and whole. So in this passage we may detect a tacit reference to the very beginning of Jesus' story - to the hour when John the Baptist said: "Behold, the Lamb of God, who takes away the sin of the world!" (Jn 1:29) [my note: Ecce, Agnus Dei!] Those words, which were inevitably obscure at the time as a mysterious prophecy of things to come, are now a reality. Jesus is the Lamb chosen by God himself. On the Cross he takes upon himself the sins of the world, and he wipes them away." (pg. 224)
Have a blessed and holy Good Friday, and may this Easter bring to you abundant grace.