Christian accounts of the Eucharist can be laid out as a series of answers to one question: in what sense is Christ present?
Transubstantiation, the Catholic position, says the substance changes and the appearances remain. Eastern Orthodoxy affirms an equally objective change, calls the elements the body and blood without qualification, and has generally refused to specify how, treating the Latin analysis as an intrusion of philosophy into mystery; the Greek term metousiosis appears in later confessions but carries less theoretical weight.
Sacramental union, the Lutheran position, holds that Christ's body and blood are truly present in, with and under bread and wine that remain bread and wine. Luther grounded it in the plain words this is my body and in his conviction that Christ's humanity shares the attributes of his divinity and can therefore be present anywhere. He refused to explain the mechanism and rejected transubstantiation as speculation. The label consubstantiation is a later coinage that Lutheran theologians usually disown.
Spiritual or pneumatic presence, the Reformed position associated with Calvin, holds that the bread and wine remain what they are and that the believer, by the work of the Holy Spirit, is genuinely fed on Christ, who is in heaven. It is a real communion with a real Christ, mediated rather than local.
Memorialism, associated with Zwingli and dominant among Baptists, Anabaptists and much of evangelicalism, treats the rite as a remembrance and a proclamation, with the bread signifying rather than conveying.
The division is old and consequential. Luther and Zwingli met at Marburg in 1529 and agreed on fourteen articles out of fifteen; the fifteenth was this one, and the Reformation split there and did not reunite. Because eucharistic teaching governs who may receive at whose table, the disagreement is also the most visible practical barrier between the churches to this day.