What is an "Elder" in The Church of Jesus Christ of Latter-Day Saints?

An "Elder" is an office in the priesthood and a title given to male missionaries as well as General Authorities over the Church. General Authorities govern the entire church or areas of church membership.

The following is information from lds.org:

Ministry


The work of the ministry is to do the work of the Lord on the earth—to represent the Lord among the people, preach the gospel, and administer the ordinances thereof. The chosen servants and appointed officers in the Church of Jesus Christ are put on earth by Him to conduct the work necessary for the salvation of mankind. A minister “called of God, as was Aaron” (Heb. 5:4), and endowed with the holy priesthood, represents the Lord when he is performing his official duties and is the Lord’s agent. Therefore, what he does “according to the will of the Lord is the Lord’s business” (D&C 64:29). The Lord has given apostles, prophets, evangelists, high priests, seventies, elders, bishops, priests, teachers, deacons, helps, and governments “for the perfecting of the saints, for the work of the ministry, for the edifying of the body of Christ” (the Church), until all have reached the spiritual stature of Christ, the whole body being “fitly joined together” by that which every part supplieth. See 1 Cor. 12:12–28Eph. 4:11–16D&C 20107. Other references are Acts 11:3014:2315:616:420:17, 2821:18Philip. 1:11 Tim. 3:8–12James 5:14.

Elders

The term elders is used in various ways in the Bible. In many instances in the Old Testament it has reference to the older men in a tribe, usually entrusted with the governmental affairs. Their age and experience made their counsel sought often. This was not necessarily a priesthood calling. Gen. 50:7Ruth 4:2Matt. 15:2, and Acts 4:5 are examples of this usage.
There were ordained elders in the Melchizedek Priesthood in Old Testament times, as in Ex. 24:9–11 and Num. 11:16. Among the Nephites there were also ordained elders in the priesthood (Alma 4:7, 166:1Moro. 3:14:16:1, 7). In the New Testament, elders are mentioned as priesthood offices in the church (Acts 14:231 Tim. 5:1, 17, 19;James 5:14–15). The term elder as used in the New Testament is from the Greek presbyteros. The detailed duties of the ordained elders in the Church today have been defined by latter-day revelation (D&C 20:42–4542:44–5246:2;107:12).
Elder is the proper title given to all holders of the Melchizedek Priesthood. Thus an Apostle is an elder in this sense, and it is proper to speak of members of the Quorum of the Twelve or the Quorums of the Seventy by this title (D&C 20:38; see also 1 Pet. 5:12 Jn. 1:13 Jn. 1:1).