The “Novus Ordo” and the “Ordinary Form” of the Mass are essentially referring to the same thing. This is the form of the Mass most people today are most familiar with. It was promulgated (originated, put forth?) by Pope Paul VI around 1969 or 1970. It permitted (did not require) that parts of the Mass may be celebrated in the vernacular (the language of the people, in our case English). But it can also be celebrated fully or partly in Latin, the official language of the Roman Catholic Church.
The “Extraordinary Form” of the Mass is the same as the Traditional Latin Mass (TLM), which was celebrated prior to the reforms of Vatican II set forth by Pope Paul VI. That is what you were witnessing today.
There are other differences between the two forms of the Mass besides the option of using the vernacular. The wording of some of the prayers is different. The cycle of readings is different. The Ordinary Form has three readings on Sunday, where the Extraordinary Form has only two. The Ordinary Form has a three-year cycle of readings, allowing for a greater variety of Scripture readings than the Extraordinary Form which has a one-year cycle.
If you attend the Extraordinary Form of the Mass, you would receive Communion kneeling and on the tongue, and there would be no Extraordinary Ministers of Holy Communion, nor would there be any female altar servers. Any hymns sung DURING the Mass (Offertory and Communion hymns) would be sung in Latin. In some churches English hymns are permitted for the processional and recessional (technically outside the Mass), although some people prefer Latin even for these.
Some people prefer the Ordinary Form because they are unfamiliar with Latin and prefer to pray in their own language. Others prefer the Extraordinary Form because of the sense of reverence and awe and mystery, and are comfortable with using a language that is reserved for holy things.
Both forms are the very same re-presentation of Christ’s sacrifice at Calvary.