The Eucharist is probably my most favourite Catholic "thing" to talk and teach about. Why? Because it is - as our Catechism of the Catholic Church says - the "source and summit of the Christian life" (par. 1324). But ave you ever been in a discussion with a non-Catholic Christian on the topic of the Eucharist and have never been able to tell them where in the Bible the evidence for this is? I mean, you're a good Catholic and all; you know what the bible pretty much says about the Eucharist but locating it is another matter. I'm about to help you with those location and some.
The first thing that needs to be considered before we dive into scripture, however, is the beginning: the birth of Christ, namely his birth place, Bethlehem. When translated from Hebrew, the name "Bethlehem" literally means "house of bread" ("beth" meaning "house", and "lehem" meaning "bread").The Hebrew "lehem" is very similar to the Arabic and Maltese word for meat/flesh, "laham", so "Bethlehem" could easily mean "house of flesh" as well, but since we are looking at the Hebrew word of a Hebrew town and with Christ's discourse in John 6 in mind, it is very fitting that Christ be born in a place named "house of bread".
Secondary to the birthplace of Christ, there is the manner in which he is born. Christ is born and as we know he is laid in a manger to rest. The manger was the feed trough for the animals. The word "manger" is very similar to the Italian word "mangiare" which means "to eat". So here we have the infant king, Jesus Christ, born in the "house of bread" and laid in a manger - an feeding trough - offered to humanity as food for the entire human race. But to fully grasp this concept of Christ being "food for the world" we have to look now to the Biblical evidence, and this is where it all comes together (parts bolded for emphasis):
 Jesus said to them, "I am the bread of life; he who comes to me shall not hunger, and he who believes in me shall never thirst.
 But I said to you that you have seen me and yet do not believe.
 All that the Father gives me will come to me; and him who comes to me I will not cast out.
 For I have come down from heaven, not to do my own will, but the will of him who sent me;
 and this is the will of him who sent me, that I should lose nothing of all that he has given me, but raise it up at the last day.
 For this is the will of my Father, that every one who sees the Son and believes in him should have eternal life; and I will raise him up at the last day."
 The Jews then murmured at him, because he said, "I am the bread which came down from heaven."
 They said, "Is not this Jesus, the son of Joseph, whose father and mother we know? How does he now say, `I have come down from heaven'?"
 Jesus answered them, "Do not murmur among yourselves.
 No one can come to me unless the Father who sent me draws him; and I will raise him up at the last day.
 It is written in the prophets, `And they shall all be taught by God.' Every one who has heard and learned from the Father comes to me.
 Not that any one has seen the Father except him who is from God; he has seen the Father.
 Truly, truly, I say to you, he who believes has eternal life.
 I am the bread of life.  Your fathers ate the manna in the wilderness, and they died.
 This is the bread which comes down from heaven, that a man may eat of it and not die.
 I am the living bread which came down from heaven; if any one eats of this bread, he will live for ever; and the bread which I shall give for the life of the world is my flesh."
 The Jews then disputed among themselves, saying, "How can this man give us his flesh to eat?"
 So Jesus said to them, "Truly, truly, I say to you, unless you eat the flesh of the Son of man and drink his blood, you have no life in you;So as you can see this caused a bit of controversy, so much so that many of Christ's followers left him at that point because they understood him on a literal level, to "eat his flesh and drink his blood". It's very important to note that while Christ was in the habit of using hyperboles and speaking in parables, he often had to explain what he meant when he spoke this way. What is most poignant about John 6 with many of his followers leaving him is that Jesus did nothing to stop them. If Jesus was being figurative and not speaking in literal terms, don't you think he would have tried to stop those from walking away by saying something to the effect of, "Hey guys, don't go! Let me explain to you what I really meant by all that!" But he didn't. Jesus was perfectly content to let them walk away because - as we know - today still people refuse to accept this "difficult" teaching and walk away from Christ in this respect.
 he who eats my flesh and drinks my blood has eternal life, and I will raise him up at the last day.
 For my flesh is food indeed, and my blood is drink indeed.
 He who eats my flesh and drinks my blood abides in me, and I in him.
 As the living Father sent me, and I live because of the Father, so he who eats me will live because of me.
 This is the bread which came down from heaven, not such as the fathers ate and died; he who eats this bread will live for ever." - John 6:35-58
So what's the Biblical evidence for the Eucharist? Let's go through a few:
Exodus 16 & Numbers 11 -- the "manna", i.e. the bread that fell from Heaven which Jesus speaks of in John 6
Matthew 26:26 (Mark 14:22; Luke 22:19) -- the sacrament of the Eucharist is instituted
1 Corinthians 10:16 -- the Eucharist is participation in Christ's body and blood
1 Corinthians 11:23-29 -- Receiving the bread and the wine in an unworthy manner is to profane against the body and blood of Christ
Exodus 12:8, 46 -- the Paschal (Passover) lamb had to be eaten
John 1:29 -- Jesus is referred to as "the Lamb of God..." by St. John the Baptist
1 Corinthians 5:7 -- Jesus is called the "paschal lamb who has been sacrificed"
And if you really want to knock an anti-Catholic's argument for six, refer to history, i.e. historical Christianity. The Early Christians celebrated the Eucharist! We turn to the wisdom and teachings of the Early Church Fathers for more proof that Christ is truly present in consecrated bread and consecrated wine:
"[heretics] abstain from Eucharist and from prayer because they do not confess that the Eucharist is the flesh of our Saviour Jesus Christ..." - St. Ignatius of Antioch, Letter to Smyrnaeans 6, 2, 2 (~110AD)
"For not as common bread and common drink do we receive these; but in like manner as Jesus Christ our Saviour, having been made flesh and blood for our salvation, so likewise have we been taught that the food which is blessed by the prayer of His word, and from which our blood and flesh by transmutation are nourished, is the flesh and blood of that Jesus who was made flesh." - St. Justin Martyr, First Apology, 66 (110-165AD)