What the veil was
The Temple in Jerusalem had an inner room called the Holy of Holies. That was where God's presence dwelt. A curtain, the veil, hung across the doorway, and it was not for decoration. It was a wall made of cloth.
God told Moses:
"Speak to Aaron thy brother, that he enter not at all into the sanctuary, which is within the veil, before the propitiatory, with which the ark is covered, lest he die"
Leviticus 16:2
Only one man could go in, the high priest, and only on one day of the year:
"But into the second, the high priest alone, once a year: not without blood"
Hebrews 9:7
Everyone else was kept out. Every ordinary person, every foreigner, every priest but one. The veil said: this far, and no further.
How big it was
Jewish tradition, written down in the Mishnah, describes the veil as a handbreadth thick, 40 cubits long and 20 cubits wide. That's roughly 18 metres by 9 metres of heavy woven cloth. It took 300 priests to carry it.
This was not a curtain anyone could rip by hand.
The moment
"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."
Matthew 27:50 to 51
It happened the moment he died, and Mark and Luke record it too.
From the top
Look at the direction. A man tearing a curtain starts at the bottom, where his hands can reach. This one tore from the top, around 18 metres up, down to the floor.
It wasn't torn open from our side. It was torn open from God's side.
What it means
The Bible explains it itself:
"Having therefore, brethren, a confidence in the entering into the holies by the blood of Christ; A new and living way which he hath dedicated for us through the veil, that is to say, his flesh"
Hebrews 10:19 to 20
The veil was a picture of his body. When his body was torn on the cross, the curtain tore with it. The way into God's presence, shut for centuries, was open.
Why this is a big deal
Before: one man, one day a year, carrying blood, behind a curtain no one else could pass.
After: anyone, any day, through Jesus.
The barrier that stood between God and man for centuries came down at the exact moment Jesus died, and it came down from above.