I think I can add a couple of drachmas' worth here. I know little of the Catholic service and the role of antiphons there, but I do know their role in the Orthodox one and have indeed sung the Greek ones in church. They are indeed short verses or even single sentences sung as hymns. The key is in the (Greek word) "antiphon", meaning "singing against" or "counter singing". They alternate indeed, as J.D.Billett summises above, but between the priest and the choir (who represent the populace). In crude translation such antiphons are for example:

priest: For it is due to you every glory, honour and (bow?), to the Father, the Son and the Holy Spirit...
choir: (1st Antiphon) Through the intercedence of Mary, Saviour, save us
priest: For the state, and power, and kingdom, and the glory, are Yours, Father, Son and Holy Ghost, now and forever
choir: (2nd Antiphon)Save us, Son of God, the one who rose from the dead, as we sing to you Halleluia(repeated 3 times)