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Anglican and Episcopalian Churches
"Eucharistic Minister", or more properly "Lay Eucharistic Minister LEM", is used to denote a lay person who assists the priest in administering the sacraments of holy communion, the concentrated bread and wine. They may also take the sacraments to those who are ill, or otherwise unable to attend Mass. LEMs usually vest in cassock and surplice rather than Alb.
Although the practice varies from Diocese to Diocese...there being no Pope in the Anglican communion...in general LEMs are recommended by the parish priest to the Bishop of the Diocese, who grants them a three-year license to practice the ministry. This license is often renewed at the priest's discretion for subsequent three year terms.
Roman Catholic Church
"Eucharistic Minister" properly refers only to an ordained priest. It is not supposed to be used in reference to any lay persons. That is specified in the Vatican document "Instruction Redemptionis Sacramentum". RS Section  states: "the only minister who can confect the Sacrament of the Eucharist in persona Christi is a validly ordained Priest. Hence the name “minister of the Eucharist” belongs properly to the Priest alone."
"Eucharist", in the Catholic Church, refers to both the celebration of the Mass, that is, the Eucharist liturgy, and to the bread and wine which are transubstantiated (changed in substance) into the body and blood of Jesus Christ. The term "Holy Communion" refers only to the latter. Thus, the terms "Eucharist" and "Holy Communion" are not interchangeable.
The distribution of Holy Communion is done by the priest, often with one or more assistant ministers, depending on the size of the service. There are two types of assistants: ordained ministers, and non-ordained lay ministers. An "Ordinary Minister of Holy Communion" is an ordained Bishop, Priest, or Deacon. See Section  of "Redemptionis Sacramentum". The term "Ordinary" refers to the fact that ordained ministers are the ones who would ordinarily assist in the distribution of Holy Communion. However, the shortage of ordained ministers has led to the need to sometimes enlist lay persons in this ministry.
An "Extraordinary Minister of Holy Communion" is a Catholic lay person who assists in the distribution of Holy Communion, either during the Mass, or afterwards, such as to housebound persons. Section  prohibits the use of other titles for these lay persons. The term "Eucharist" does not appear in their title, because it would imply a broader function, which is reserved for the priest alone. Despite these instruction, a very common error is the use of the terms "Eucharistic Ministers", or "Extraordinary Ministers of Eucharist. These improper titles have appeared on several book titles, instructional handbooks, and in many articles. Likewise, erroneous abbreviations, such as "EM", "EME", or "EEM", are commonly used instead of the proper abbreviation "EMHC".
Section  provides instructions on the proper use of EMHC: "the extraordinary minister of Holy Communion may administer Communion only when the Priest and Deacon are lacking, when the Priest is prevented by weakness or advanced age or some other genuine reason, or when the number of faithful coming to Communion is so great that the very celebration of Mass would be unduly prolonged." However, many parishes do not follow these instructions. Some routinely use EHMCs when there is no lacking of Priests and Deacons, or they employ an large number of EMHCs, reducing the Communion distribution to a few minutes. One factor is lack of knowledge of the Vatican instructions. Another is the popularity of participating in this service.
Vatican document "Instruction Redemptionis Sacramentum" sections 154 through 158: http://www.vatican.va/roman_curia/congregations/ccdds/documents/rc_con_ccdds_doc_20040423_redemptionis-sacramentum_en.html