Supersessionism refers to the concept that the New Testament supersedes the Old Testament. But it can also refer to other vaguely related ideas that the Church supersedes ancient Israel, that the Church fulfills the missions Israel held, or that Christianity brings something new.
The term developed in Protestant scholarship in the 1970's-1980's, and its normal use is to portray traditional Church teachings on these questions, often in a negative light.
"Supersession" as a simple concept
"Supersession" comes from the word "supersede", which means “to set above; to make void or inoperative by a superior authority; to stay, suspend or supplant”  It can also mean to overrule, override, or replace the function of something.
For example, an amendment to a law supersedes the original law, and a governor's pardon overrules a sentence prescribed by law, and a remodeled home or car replaces the old one. Note that in each of these cases that which is superseded may remain in existence fully or partially.
"Supersession" in Orthodoxy
Orthodox writers on occasion use the term "supersession" to describe the relationship between the Old and New Testaments or between certain ideas in them. According to the Orthodox Study Bible, an Orthodox priest is not merely reenacting the ancient Israelite priesthood, but is "a minister of a new covenant that supersedes the old".
Fr. Evan Armatas of Saint Spyridon Greek Orthodox Church, uses it to describe the hierarchy of importance of parts of the Bible:
- The New Testament, as we call it, is the last part of the Christian Bible, and we accept both Old and New, although we do believe that the New Testament supersedes the Old... In the Church, we keep the hierarchy of the Bible by the way we do it liturgically. Where is the Gospel? On the altar table. Where [are] the epistles and the Old Testament? Out on the side. 
Pope St. Leo the Great in his Sermon on the Passion described Old Testament elements ceasing, or passing into or changing into New Testament ones, noting that "the True Sheep had to supersede the sheep which was its antitype". The idea of Typology in the Old Testament prophesying the new one is an important concept in Orthodox theology.
The Church father Tertullian commented: "the Creator indeed promised that "the ancient things should pass away,"(Is 43:18-19, 65:17; II Cor 5:17) to be superseded by a new course of things which should arise, whilst Christ marks the period of the separation." Tertullian commented on the Apostle Paul's words that God "called you to His grace to another gospel"(Gal 1:6-7) by explaining that St. Paul means “'another' as to the conduct it prescribes, not in respect of its worship; … because it is the office of Christ’s gospel to call men from the law to grace."
Clarifying the Church's use of "Supersession"
Despite the New Testament's precedence, and despite certain Old Testament ritual elements ceasing or changing, the Old Testament continues to have importance: It remains an important source of learning, as St. Paul writes: “All scripture is given by inspiration of God, and is profitable for doctrine and for instruction in righteousness”. (II Tim 3:16)
Maximus the Confessor sees the two Testaments as complementary, writing: “The Old Testament provides to the knowledgeable man the modes of virtues. The New Testament gives the practical man the words of true knowledge.”
As to the relationship between ancient Israel and the Church, there is a continuation between the two as St. Paul described it in Romans 11. There he portrayed Israel as a spiritual community from whom some branches had been broken off, while others (gentile Christians) had been grafted in, while keeping the hope that the broken branches would return.
Criticisms of the term "Supersession"
While the term might describe certain Orthodox views, it is uncommon among Orthodox worldwide, since an exact translation does not exist in Slavic languages. It is also rare in patristic writings.
The term may cause confusion because supersession can refer to a new thing adding onto an older thing that still remains (eg. adding a new provision onto a law), or to an older thing being destroyed in every sense (eg. a law that has been canceled).
- Kendrick Kinney: A Law Dictionary and Glossary, 1893, Callaghan and Company, p. 642
- Orthodox Study Bible, Conciliar Press, p. 1635
- Fr. Evan Armatas, "Formation of the New Testament Canon", Ancient Faith Radio. http://ancientfaith.com/podcasts/transforminglives/formation_of_the_new_testament_canon
- St. Leo the Great, On the Passion, VII. http://www.newadvent.org/fathers/360358.htm
- Tertullian, Against Marcion, Book 3, Chapter 2. http://st-takla.org/books/en/ecf/003/0030448.html
- St Maximus the Confessor, Exegesis of Zechariah 4:1–3