The celebration of the Mass can be still said in Latin, as it is the official language of the Church.
In many oratories, shrines, chapels, and even parishes, Masses continue to be celebrated in this language. In fact, the Second Vatican Council encouraged the use of Latin. Latin hymns and prayers were to be retained even in Masses during which the vernacular languages are used.
Mass was first celebrated in the language of our Lord, the ancient Aramaic. When Peter, the first pope, went to Rome, the center of the Empire, Greek was the language used by scholars, and it then became the language of choice for many centuries in the Liturgy. Theology and philosophy were all taught in this language. Latin was considered the common language of the people. The theological opinion of the time said that Divine Liturgy, the Mass, should use the language of scholars, which was a sign of dignity.
After the Peace of Constantine, the Church became legitimate and expanded tremendously. Roughly, this coincided with the fall of the Roman Empire in the West and the development of the vernacular languages. Latin then became the language of scholars, theologians, and philosophers, and so the Mass was then translated into it. In the Eastern Empire, which was centered in Constantinople, Greek was retained.
For centuries, Latin was commonly used in Church documents, Canon Law, reading of scripture, and the celebration of the seven sacraments and prayers. Latin was and is the official language of the Church. It was only in recent times that the Holy See allowed the vernacular languages to be used in its celebration of the Sacraments.
However, this usage in no way diminishes the importance of Latin in the Liturgy.