What exactly does the Church mean by tradition?

We must distinguish divine Tradition from mere ecclesiastical tradition or custom. Divine Tradition comes from God, either through the written word of the Bible or through the oral teaching of Christ himself or his apostles. Because it is revealed by God, divine Tradition may not be altered by men.

Ecclesiastical tradition or custom, on the other hand, originates with the Church’s pastoral and disciplinary authority and may change.

The example you mentioned, Mass in the vernacular, falls into the category of ecclesiastical tradition. The first Mass, the Last Supper, was probably in Aramaic–possibly in Hebrew. As Christianity spread into the pagan world, the liturgy was translated into Greek and Latin (the vernacular languages of that day). Only with the passage of time and the abandonment of these languages as universal tongues did they take on the quality of sacred languages.

In the Western Church, Latin continued to be the common language of the liturgy until recently. In the Eastern churches in communion with the Catholic Church, Greek and other languages, including Aramaic, are used. The Mass in English isn’t an alteration of divine Tradition, but of ecclesiastical custom.