What does the Bible say about the Eucharist?

First of all, what is the Eucharist?  

The Eucharist is the Sacrament of the Bread and Wine which are the Body and Blood of Jesus Christ.  

The word, "eucharist" comes from the Greek word, "eucharistia" which means "thanksgiving".  

The reason that this word has come to represent the Sacrament of the Body and Blood of Christ is because in Luke 22:17, it is recorded that in the Last Supper, when Jesus established this Sacrament,  "…He took the cup, and gave thanks, and said, Take this, and divide it among yourselves".

And likewise in Luke 22:19, "He took bread, and gave thanks, and brake it, and gave unto them, saying, This is my body which is given for you: this do in remembrance of me."

The underlying word for "gave thanks" is "eucharistia.  

It is used in the same way by St. Paul in 1 Cor 11:24.

1 Corinthians 11:24

24 And when he had given thanks, he brake it, and said, Take, eat: this is my body, which is broken for you: this do in remembrance of me.