Within the Catholic Church, there are various ways of celebrating the Eucharistic Sacrifice, the other sacraments, and sacramentals. These  “forms” or  “uses” are distinct expressions of the single Catholic faith through different ways of carrying out the liturgy. These forms and uses often involve distinct sacred languages, gestures, rituals, etc. These differences are most obvious when we compare the Roman Rite of the Latin Church with various liturgical rites of Eastern Catholic Churches. Nevertheless, even within the Latin Church, there is a diversity of ways of expressing the Roman Rite.

In 2007, Pope Benedict XVI established that:

"The last version of the Missale Romanum [Roman Missal] prior to the Council, which was published with the authority of Pope John XXIII in 1962 and used during the Council, will now be able to be used as a Forma extraordinaria [extraordinary form] of the liturgical celebration.  It is not appropriate to speak of these two versions of the Roman Missal as if they were “two Rites”.  Rather, it is a matter of a twofold use of one and the same rite." (Letter accompanying Summorum Pontificum).

In other words, the form of celebrating Mass, the sacraments, and sacramentals as it existed in the Latin Church in 1962, prior to the reforms that followed the Second Vatican Council, is an “extraordinary form” of the Roman Rite, complementing the ordinary form, which is the form of celebrating these rites as revised following the Second Vatican Council. This extraordinary form is also sometimes called the usus antiquior (older use), the Traditional Latin Mass (TLM), or the Tridentine Mass.

Pope Benedict XVI explained it, briefly, this way:

"There is no contradiction between the two editions of the Roman Missal.  In the history of the liturgy there is growth and progress, but no rupture.  What earlier generations held as sacred, remains sacred and great for us too, and it cannot be all of a sudden entirely forbidden or even considered harmful.  It behooves all of us to preserve the riches which have developed in the Church’s faith and prayer, and to give them their proper place." (Letter accompanying Summorum Pontificum)

In other words, the way of celebrating the Roman liturgy prior to the Second Vatican Council is not a threat to the way of celebrating it in accord with the liturgical books promulgated after the Council. Rather, it is a good, true, and beautiful part of our heritage as Roman Rite Catholics, worth preserving and valuable for the worship of God and growth in sanctity today as it has been over the centuries.

There are many videos of extraordinary form Masses (and Baptisms, etc.) online. Some of the Masses are Low Masses (a priest celebrates the Mass and does not sing); some are Missae cantatae or “sung Masses” or “high Masses” (a priest celebrates and sings the Mass); and some are Missae solemnes or “solemn Masses” or “solemn high Masses” (a priest celebrates, assisted by deacon and subdeacon, the Mass is sung).

Here's a video of a solemn Mass in Sacrament, CA.

Generally speaking, any priest of the Latin Church who may celebrate Mass and other sacraments legitimately in the ordinary form may also celebrate them in the extraordinary form if he knows how to do so. No special permission is required. (See Summorum Pontificum, a. 2; and Universae Ecclesiae, n. 20)