OF is the Ordinary Form of the Roman Rite. It is the same thing as the NO - which means the Novus Ordo (Latin for New Order). It is sometimes called the Pauline Mass, the Mass of Pope Paul VI, or Mass according to the 1970 Roman Missal.
EF is the Extraordinary Form of the Roman Rite. It is the same thing as the TLM (Traditional Latin Mass) or the Tridentine Mass.
It is also sometimes referred to as the Gregorian Rite, the usus antiquior - the “ancient use” of Mass, the Mass of Blessed Pope Jonn XXIII, the Mass of Pope Saint Pius V, or Mass according to the 1962 Roman Missal.
You can compare the 2 side by side
here.
Most parishes offer the OF. It became the normative Mass for Roman Catholics in 1970 (actually the 1st Sunday of Advent 1969). The official language of this Mass is Latin. But, there are official translations in many, many different languages.
Even though, according to the 2nd Vatican Council, Latin was supposed to be retained in the Mass after vernacular languages were permitted, it is incredibly difficult (and in many places impossible) to find any Latin used at all. This Mass is known to many people in the U.S. as the “English Mass”.
Odds are that if you stop into an unfamiliar Catholic parish this will be the Mass used. It could be in English, Spanish, Latin (highly unlikely) or any number of other languages.
The EF on the other hand is always celebrated in Latin. The sermon will typically given in English (or Spanish, or whatever the primary language of the people happens to be). It is also permissible for the Readings to be in the vernacular. More commonly, however, is for the priest to read them in Latin and then reread them in another language at the start of the sermon.
The EF Mass is the Mass familiar to most Roman Catholics for many centuries. The nickname “Tridentine Mass” implies a connection to the Council of Trent. But it is really much older than that - back to the time of Pope Gregory the Great (hence the name Gregorian Rite).
After the introduction of the Novus Ordo Mass it was only permitted to be said with the permission of the local bishop until 2007 when Pope Benedict XVI released his Motu Proprio
Summorum Pontificum. He made it clear that the Traditional Latin Mass had never been legally suppressed and therefore any Roman Catholic priest could offer it.
**The Roman Missal promulgated by Paul VI is the ordinary expression of the ‘Lex orandi’ (Law of prayer) of the Catholic Church of the Latin rite. Nonetheless, the Roman Missal promulgated by St. Pius V and reissued by Bl. John XXIII is to be considered as an extraordinary expression of that same ‘Lex orandi,’ and must be given due honour for its venerable and ancient usage. These two expressions of the Church’s Lex orandi will in no any way lead to a division in the Church’s ‘Lex credendi’ (Law of belief). They are, in fact two usages of the one Roman rite.
It is, therefore, permissible to celebrate the Sacrifice of the Mass following the typical edition of the Roman Missal promulgated by Bl. John XXIII in 1962 and never abrogated, as an extraordinary form of the Liturgy of the Church.**
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(Summorum Pontificum, Article 1)
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**In Masses celebrated without the people, each Catholic priest of the Latin rite, whether secular or regular, may use the Roman Missal published by Bl. Pope John XXIII in 1962, or the Roman Missal promulgated by Pope Paul VI in 1970, and may do so on any day with the exception of the Easter Triduum. For such celebrations, with either one Missal or the other, the priest has no need for permission from the Apostolic See or from his Ordinary.
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(Summorum Pontificum, Article 2)
*Does this answer your questions?
James