The Traditional Latin Rite refers to the Mass that was the norm prior to the promulgation of the Novus Ordo Missal in 1969. Some think that it has its origins in the Council of Trent, but in fact the rite is far older than the Council of Trent. It is quite possible that the rite has its origins in St Peter.
The Traditional Latin Mass is rich in Catholic Tradition. These include some of the most beautiful chants in the treasury of the Church. The traditional Latin Mass also more fully represents the sacrificial aspect of Holy Mass which is the true nature of the Mass. The understanding of the priest as the person set aside to offer sacrifice to Almighty God in expiation for the sins of the faithful.
Silence is also a big difference between the two rites. Meditation is fostered through the silence and chants of the Traditional Latin Mass. This also allows a Catholic to more effectively pray the Mass. Silence also allows one to place oneself on Mount Calvary during the Mass.
While there are similarities between the Novus Ordo and the Traditional Latin Mass they are in fact quite different rites. The Traditional Latin Mass has a hierarchical structure of Masses of varying solemnity, ranging from a Low Mass to a Solemn Pontifical Mass. This form of the Mass also has more rigorous rubrics.
More recently the Holy Father, Pope Benedict XVI called the Traditional Latin Mass the extraordinary form. He issued a Moto Proprio, called: Summorum Pontificum. This allowed for a greater use of this rite (according to the 1962 Missal).
The video above is a commentary on a Low Mass by the Priestly Fraternity of Saint Peter, who exclusively celebrate this form of the Mass.