The foundation of all Kontakia (note the plural) is the Kontakion of St. Romanos the Melodist. According to established scholars, the form of the Byzantine �kontakion� was probably based in Syriac hymnographical traditions and was often described as a �sermon in verse accompanied by music�.
The entire Kontakion is rarely sung today in Eastern services -- the only exception to this Akathist Hymn which is sung throughout the Fridays of Lent in Greek (and some Russian) traditions. You will see in Orthros, that the first Kontakion and Eikos have been retained in ordinary services and are read after the 6th Ode.
What we sing in Church during the Divine Liturgy can be the first ode of the total Kontakion (which originally consisted of 18 - 24 stanzas). The best example of this is the Kontakion to the Mother of God (To You, O Champion Leader) which is sung in the Greek traditions during Lent. (Of course, it is also true that there are "independent" Kontakia which, for our purposes here, is simply a Troparion by another name. Although, I would note, that if you look and compare individual Troparia and Kontakia you will notice a difference (often subtle) in emphasis and theological reflection.)
The Greek and Russian traditions have tended to deal with Kontakia in the Divine Liturgy differently. The Greek typicon prescribes general Kontakia for seasons of the year. During "normal" time, the Greeks sing "Steadfast Protecress" -- and, as I have already mentioned, during Lent they sing "O Champion Leader." The Russian typikon on the other hand prescribes the singing of the Kontakion of the Resurrection each Sunday and often the singing of an individual Kontakion for the saint of the day.
The best historical discussion of this issue can be found at
http://www.liturgica.com/html/litEOLitMusDev1.jsp?hostname=null
