1. Real Presence. This belief is that somehow (and there are a variety of theories) the bread and wine either become or are the literal body and blood of Christ. I believe these views are unsupported by scripture, and come from those who do not understand that the Lord’s physical body is in heaven with the Father, and that He is only on earth in a spiritual sense with individuals (Matt. 28:20), and collectively where “two or three are gathered” unto His name (Matt. 18:20).
   • Transubstantiation. This is the view that during the ceremony the bread is changed from the substance of bread into the literal flesh of Christ’s body, and that the wine is changed into the literal substance of Christ’s blood. This view is held by the Roman Catholic Church and many Orthodox Churches. The word describes the belief; “trans” (change) “substantiation” (substance). The Orthodox Church believe essentially the same thing, although they do not attempt to describe how the change occurs as the Romans do, but rather view it as a mystery.
     

     “…By the consecration of the bread and wine there takes place a change of the whole substance of the bread into the substance of the body of Christ our Lord and of the whole substance of the wine into the substance of his blood. This change the holy Catholic Church has fittingly and properly called transubstantiation.”

     – From the Council of Trent (1545-1563)

     This continual reenactment of the death of Christ may be contemplated in the name “Thyatira” which means ‘continual sacrifice’ (Rev. 2:18), the “continual sacrifice” referring to the Catholic mass.
   • Holy Mystery. This is the view that somehow the bread and wine are the body and blood of Christ, but that it is impossible for man to understand how the change occurs, or if there is a substantive change; rather it is a “holy mystery”. This view is held by main-line Anglicans.
   • Consubstantiation. This is the view that the bread and wine, rather than change substance, they retain their original substance, and the substances of the literal body and blood of Christ are added to them, respectively. Hence the prefix “con” or “with”. Therefore, a person partakes of both the bread and wine, as well as the body and blood of Christ. This view is held by some Anglicans.
   • Sacremental Union. This is the view that when Jesus instituted His supper, and said “this is my body” and “this is my blood” referring to the bread and wine, he forever united the literal body and blood with the bread and wine. Therefore, a person partakes of both the bread and wine, as well as the body and blood of Christ. Christ is mysteriously present “in, with, and under” the bread and wine. This view is held by Lutherans. They reason like this; Christ is a man with a physical body, Christ is also God who is omnipresent, therefore the body and blood of Christ are everywhere including the bread and wine.
   • Pneumatic Presence. This is the view that the literal body and blood of Christ are present through the Holy Spirit. The literal body and blood are not physically ingested by the person, but they are “mystically received” by the person. This view is also known as the Real Spiritual Presence. It is held by Reformed Baptists, Evangelical Anglicans, Presbyterians, and other Reformed Churches.
2. Memorialism or Zwinglianism. This view is that the bread and wine are symbols only and represent the literal body and blood of Christ. The body and blood are in no way literally present during the Lord’s Supper. Rather than a “reenactment” of the cross, the Lord’s Supper is a memorial of Christ in which His death is “shown forth”. This view is held by Brethren, Baptists, and Anabaptists. I believe this is the scriptural position. This view was famously held by Ulrich Zwingli, who went head-to-head with Martin Luther on the issue of Eucharistic Theology at the Marburg Colloquy. 
3. Suspensionism. This view is that the Lord’s Supper was not intended to be kept in perpetuity, or that it was never intended to be literally done as an outward ordinance. This view is held by Quakers and the Salvation Army. This position is clearly wrong, because the Lord said “This do… until I come” (1 Cor. 11:23-26).

“With tears in his eyes, Zwingli approached Luther , and held out the hand of brotherhood, but Luther declined it, saying again ‘Yours is a different spirit from ours.’ Zwingli thought that differences in non-essentials, with unity in essentials, did not forbid Christian brotherhood. ‘Let us,’ he said, ‘confess our union in all things in which we agree; and, as for the rest, let us remember that we are brethren. There will never be peace in the Churches if we cannot bear differences on secondary points.’ Luther deemed the corporal presence a fundamental article, and construed Zwingli’s liberality into indifference to truth. ‘I am astonished,’ he said, ‘that you wish to consider me as your brother. It shows clearly that you do not attach much importance to your doctrine.’ … Turning to the Swiss, the Wittenburgers said, ‘You do not belong to the communion of the Christian Church. We cannot acknowledge you as brethren.’ They were willing, however to include them in that universal charity which we owe our enemies.”

— Phillip Schaff, History of the Christian Church, Volume 7, Chapter 5, Sections 107-108