"Real presence" is the term used to describe the Christian belief that Jesus Christ is in some way present in the Eucharist, the memorial of the Last Supper (also known as Holy Communion and the Lord's Supper).
There are many different views of the Eucharist, ranging from a total transformation of the Eucharistic elements of bread and wine into the body and blood of Jesus (transubstantiation) to a totally symbolic view, found largely in Evangelical, Fundamentalist, and Pentecostal circles.
Specific Eucharistic Doctrines
Roman Catholic: the Catholic Church believes in transubstantiation, the doctrine that the whole substance of the bread and wine is transformed into the whole substance of Jesus Christ--His body, blood, soul, and divinity. This church believes that the service is more than a memorial, and also that the Eucharist (and thus the Mass) is a re-presentation of Christ's sacrifice on the Cross. Those perspectives are not, however, part of the doctrine of the Real Presence.
Anglican: The Anglican churches officially believe in the Real Presence received only in a non-carnal way and explicitly reject the doctrine of transubstantiation. However, a small minority of "Anglo-Catholic" Anglicans do accept transubstantiation.
Lutheran: Luther and Lutherans today believe in the Real Presence of Christ's body and blood in the Eucharist. However, rather than holding that the whole substance of the bread and wine is transformed (i.e. transubstantiation), they consider the body and blood to exist alongside the bread and wine--"In, with, and under the forms of bread and wine." (This is often miscalled "consubstantiation").
Methodist: The United Methodist Church and most of the smaller Methodist bodies believe in the Real Presence but reject transubstantiation along with most Protestant denominations. The UMC refers to Christ's presence in the Eucharist as a "Holy Mystery," and prefers not to explain the details of Christ's presence.
Reformed/Presbyterian: Reformed and Presbyterian churches generally believe in a "pneumatic" or spiritual presence of Jesus in the Eucharist; while no actual change takes place in the bread, there is still a Real Presence of Jesus which the people experience during the celebration of the Lord's Supper.
Baptist/Evangelical/Pentecostal/Fundamentalist: While the theologies of these traditions differ from each other, their views on the ordinance (Lord's Supper) itself are generally the same. It is simply a symbol and a memorial of the Last Supper by which worshipers remember Christ's body and blood given up on the Cross. It is not a re-presentation of the sacrifice of the Cross--which is believed to have been accomplished once--or a literal sharing in Christ's physical body or blood.
Real Presence gradations of understanding within the Catholic and Orthodox traditions
The Roman Catholic doctrine of transubstantiation was formulated by the teachings of St. Thomas Aquinas during the thirteenth century. Aquinas taught that after the prayer of consecration or blessing at the Altar, the bread and wine were no longer bread and wine, though they appear to be so to the senses. His teaching is based on the distinctions that Aristotle made between essence or substance of things, and the appearance of things. Every thing has an essence and an outer manifestation which he called the "accidents". Thus. according to Aquinas, though after the prayer of "invocation" or calling on the Holy Spirit on the elements of bread and wine, these elements still taste, feel, smell, like bread and wine, they are in reality, because of the word of Christ and the Scripture, His body and blood.
The Orthodox and Aramaic Churches, as well as some modern derivatives, hold to the real presence of Christ in the Eucharist but do not subscribe to transubstantiation, or, consider transubstantiation to be only one possibility for explaining His presence. These churches formulate their position in this manner: It is clear from Scripture and the way the earliest Christians after the New Testament, that His life is imparted at the Eucharist in a real way. They most often point out that Christ said "This is my body" and "This is my blood". Further, it is pointed out that in John, Chapter six, "eat my flesh" and "drink my blood" is to be understood "eucharistically" as is shown by the non-inclusion in the Last Supper account and the "words of institution"--"This is my body..." and "This is my blood"--in the Gospel of John. By contrast, they appear in the other three Gospels. There was no need to include them as it had already been dealt with in John 6.
Some of the Orthodox and Aramaic Churches understand the Real Presence of Christ in assumptionist terms rather than by transubstantiation. The assumptionist understanding is analogous to the way the Son of God took on human flesh. When the Son of God became man, He did so, not by obliterating the humanity by His divinity but rather "taking up" or assuming humanity into Himself. Thus He became truly the Son of God and the Son of Man--a true unity of Deity and Humanity. They hold a similar view with regard to the Eucharist. After the consecration of the bread and wine, Jesus' real presence is there, His body and blood with all the attendant life, strength and grace. This has been done without the obliteration of bread and wine, but, as in the words of Ireaneaus, plain bread has become "heavenly bread" by being assumed into the more dominant flesh of the Son of God, and so also with blessed wine.