Matthew 14:22-25 (NIV):
22While they were eating, Jesus took bread, gave thanks and broke it, and gave it to his disciples, saying, "Take it; this is my body."
23Then he took the cup, gave thanks and offered it to them, and they all drank from it.
24"This is my blood of the covenant, which is poured out for many," he said to them. 25"I tell you the truth, I will not drink again of the fruit of the vine until that day when I drink it anew in the kingdom of God."
Transubstantiation is the Roman Catholic belief that, during communion, the bread and wine, upon consecration, literally become the "body" and "blood" of Jesus Christ, although the physical qualities of the materials, such as the taste and fragrance, do not appear to change; the theological belief is based on the Church's scholarly interpretation of the Biblical verses quoted above.

The doctrine of transubstantiation was first adopted by the church at the Fourth Lateran Council in 1215, and was re-affirmed in 1551 at the The Council of Trent. Sadly, it was also at this Council that a portion of the doctrine was written as such that it has developed as a major rift between Catholic and Protestant branches of the church. In particular, it was stated in Session 13, Canon 2 that:
If anyone says that in the sacred and holy sacrament of the Eucharist the substance of the bread and wine remains conjointly with the body and blood of our Lord Jesus Christ, and denies that wonderful and singular change of the whole substance of the bread into the body and the whole substance of the wine into the blood, the appearances only of bread and wine remaining, which change the Catholic Church most aptly calls transubstantiation, let him be anathema. 1

It should, at this point, be noted that the concept of transubstantiation differs from many Protestant beliefs that the bread and wine are, to varying degrees, symbolic. Thus, the significance in the above quote is that, due to this decree, non-Catholic believers are considered persona non grata at a Catholic Mass -- a decision that, in a time where we are seeing churches of various beliefs reuniting, is still an unfortunate schism that has not been resolved, and doesn't look to be, any time soon.

Works Referenced and Cited
1"Orthodoxy and Transubstantiation" :

The NIV Biblical translation, including the portions quoted above, are © Copyright 1973, 1978, 1984 by International Bible Society