Baptism and the Eucharist are two sacraments (as Catholics call them) or ordinances (as Protestants often call them) in the New Testament church that have clear types in the Old Testament.
One way to generally distinguish Catholic theology from Protestant theology is to say the following:
In Catholic theology, the New Testament sacraments are MORE miraculous than their Old Testament types.
In Protestant theology, the New Testament ordinances are LESS miraculous than their Old Testament types.
Let's consider baptism first. In the Old Testament, the types of baptism (the first creation, the story of Noah, the passing through the Red Sea, the water that came from the Rock) all constitute powerful acts of God that in most cases were powerful, visible miracles. In many cases, the types of baptism saved God's people while simultaneously washing away sinful oppressors.
For Catholics, these types all point to the even GREATER miracle that Christ performs when He baptizes someone into the family of God and washes away any stain of original or personal sin. (Note: Christ is the one who performs baptisms. Baptism is Christ's work, not ours.) At baptism, we become a new creation. At baptism, the Holy Spirit that hovers over the waters enlightens the soul and makes us children of God. Baptism is the spiritual counterpart to circumcision in the Old Testament. Through baptism, we are initiated into the family of God. Baptism is a great, miraculous event for Catholics.
For many Protestants, baptism is not a miraculous event. There is no grace attached to baptism, since baptism is strictly an outward symbol. All of those miraculous types of baptism in the Old Testament foreshadowed a New Testament ritual that is actually spiritually powerless. People may have been saved through the types of baptism in the Old Testament, but no one is saved by baptism in the New Testament. Sin may have been washed away by the Old Testament type, but the New Testament reality is powerless to wash away sin.
Now let's consider the Eucharist, the bread of the Lord's supper. In the Old Testament, the manna that was given in the desert was truly miraculous. This bread that came down from heaven fed the Israelites in their journey across the desert, and it fell in double quantities on Friday so that they Jews could truly rest from their food gathering on the Sabbath. Right before Jesus's famous bread of life discourse in John 6 (which remember, occurred at the time of Passover, one year before he was to institute the Eucharist at the last supper and give his life on the cross as the true Paschal Lamb of God), Jesus performed another miracle involving bread, in which he multiplied the loaves to feed the thousands of people that followed him.
For Catholics, these types of the New Testament Eucharist are less miraculous than the reality to which they point, in which Jesus truly feeds us and communes with us in his fullness--in his Body, Blood, Soul, and Divinity--hidden under the species of bread and wine. Since the beginning of Christianity, Christians have faithfully passed on the apostles' understanding of Jesus's words in John 6, which is that He meant what he said. Jesus insists over and over, and says in as many different ways as possible, that His flesh is real food and his blood is real drink. He even goes so far to say that this flesh is the same flesh that he will give for the life of the world. All the early fathers of Christianity, including many who learned the faith from St. John the apostle himself, all believed that the bread and wine of the Eucharist is truly Jesus Christ himself. Indeed, the Eucharist is the miracle of miracles, and the miraculous presence of Jesus in the Eucharist continues to this day to manifest itself physically to believers. (Check out this most recent, developing story, which appears to be yet another Eucharistic miracle! For other Eucharistic miracles, click here.)
For Protestants, the Lord's supper is an ordinance that imparts no grace in and of itself apart from whatever grace is involved in remembering Jesus's passion and death on the cross. The bread and the grape juice are certainly not miraculous in any way, shape, or form, and are usually discarded in the trash can after the service. This is not viewed as wrong, since the elements are not considered miraculous. Most Protestants have a difficult time explaining why Jesus would have them go through a simple ritual of eating bread and drinking grape juice as a means of remembering his passion and death, and thus, the ritual often is justified in terms of our need to be obedient. It is typically scheduled to occur once every three months, even though the members of the early church celebrated the Eucharist every time they met to worship (see Acts 2).