The prayer itself is of ancient origin. Our oldest record of it comes from the twelfth century, but the website ourcatholicprayers.com tells us:
According to The Golden Legend, a famous 13th century work about the saints, Pope St. Gregory the Great heard angels singing the first three verses from the Regina Coeli during a procession in the 6th century and was inspired to add the fourth line “Ora pro nobis deum” (“pray for to us to God” in Latin). Although this story is itself considered to be a legend, it is, as Father Herbert Thurston once put it in his book Familiar Prayers, “inseparably associated with the Regina Coeli.”
We pray it in place of the Angelus during the season of Easter, at which time it also serves as the the Marian Antiphon at the end of Compline (Night Prayer). I have posted the prayer itself, in both English and Latin, below the clip.
Queen of Heaven, rejoice, alleluia.
The Son whom you merited to bear, alleluia.
Has risen, as He said, alleluia.
Pray for us to God, alleluia.
V. Rejoice and be glad, O Virgin Mary, alleluia.
R. For the Lord has truly risen, alleluia.
Let us pray.O God, who through the resurrection of your Son Jesus Christ gave rejoicing to the world, grant, we pray, that through his Mother, the Virgin Mary, we may obtain the joy of everlasting life. Through Christ our Lord. Amen.
Regina cæli, lætare, alleluia:
R. Quia quem meruisti portare, alleluia,
Resurrexit, sicut dixit, alleluia,
R. Ora pro nobis Deum, alleluia.
Gaude et lætare, Virgo Maria, alleluia.
R. Quia surrexit Dominus vere, alleluia.
Oremus.Deus, qui per resurrectionem Filii tui, Domini nostri Iesu Christi,
mundum lætificare dignatus es:
præsta, quæsumus, ut per eius Genitricem Virginem Mariam,
perpetuæ capiamus gaudia vitæ.
Per eundem Christum Dominum nostrum. R. Amen.