Aramaic was the first language of Christian liturgy. And it was the common language of the first Christians,
who were Palestinian Jews. As Hebrew was the language of scripture and formal worship, and Christian worship occurred in the home where Aramaic was spoken.
In some of the Mediterranean world, the common language was Greek, and it became the language of liturgy in that region and remained so until the early third century.
Even Eucharist is a Greek word, which means thanksgiving. The word Kyrie eleison and the words liturgy, baptism, evangelize, martyr, and catechumen, among other familiar church words, are also Greek in origin.
Starting from third century B.C., what we call “classical” Latin was the language of the Roman aristocracy and the educated classes. During the time Jesus was born which was the period of the reign of Augustus Caesar, the language started to change. The aristocracy of Roman was destroyed by war and political infighting and when they disappeared, their language went with them. And Classical Latin was replaced by a less refined version of the language.
During the third and fourth centuries A.D. this Latin began to take over Greek as the common language of the Roman world and soon, it became the language of the liturgy.
Actually, how this change in the liturgy came about is uncertain. There were some written examples of Eucharistic Prayers, but they were models, not prescribed prayers. The last of such document in Greek was written around the year 215. As at the time of the sixth century, the Roman Canon (which is still in use, and also called Eucharistic Prayer I) appears, completely in Latin and prescribed for use exactly as it was written.
It seems that the main part of the Roman Canon was developed and used first, probably even in liturgies that were partly in Greek and partly in Latin, until the final Latin version evolved. And Christians had not used Latin for worship prior to this, words had to be adapted or imported (often from Greek) to express Christian ideas, which started the development of an ecclesiastical form of Latin.
An evidence was also shown that the Roman Canon was influenced by prayers from the Eastern churches.
Though the Latin was developed into various modern languages, Latin still remained the sole language of the Roman Rite until the Second Vatican Council returned to the original way of Christianity that people should worship in a language they understand.