Welcome to the forums!
I’ll give it my best shot.
First, note that there are more than just those forms of Mass. The Catholic Church is divided up into 22 or so different rites. A rite is basically a particular visible form of expressing the faith. The one you are thinking of is called the Roman, or Latin rite. It is the Western expression of Catholicism. In the East, there are many different rites. All of them share the same doctrinal beliefs, and all are gathered in unity under the Pope - these things are hat makes them all Catholic. What makes them different from one another is the specific way that the faith is expressed - like in the Mass, in the prayers, and sometimes in the way that theology explains the doctrinal beliefs. Ask about this if you have more questions. It’s kindof hard to explain and questions may help you get a better conept of it.
Now you are asking about the Roman rite. In the middle ages, there were a lot of different variations on how Mass was said in different places in the western Church. Finally, after the Council of Trent in 1500, a new Missal - the book that has all the prayers and instructions for how Mass is said - was put together to replace all of these so that A) there would be no question of the orthodoxy of the Mass no matter where it was said, and B) there would be unity in the Roman rite, rather than a jumble of different practices. This is the Missal that was used up until the 1960s, with a few changes and updates that Popes introduced every now and then over the centuries.
In this Missal, there were two ways of saying Mass - the High Mass, and the Low Mass. The Low Mass was a basic Mass, with the bare necessities. The High Mass was a more solemn Mass, and it had music, and all the other “smells, bells, and whistles” as they say. It also requires more servers and deacons assiting the priest saying the Mass. It was basically a more ceremonious way of saying the Mass. Think of your two different forms of graduation a college may have - one held in the middle of the year for those who graduated after the fall semester, and the other held in the spring for most of the graduates. They both do the same thing, but one has many more people involved and is much more dressed up and solemn.
In the 1960s, the Church decided to put together a new Missal with the intention of better helping those in our modern culture to experience the Mass. This Missal - called the Pauline Missal or, as you will see it very often called the Novus Ordo, which means “new order” - has been used since. The older form was not technically disallowed, but many bishops believed it was or disallowed it themselves. As a result, in 2007 Pope Benedict issued a document which declared that both the old Mass, and the Pauline Mass could be said at any time. He said that the Pauline Mass would be called the Ordinary Form of the Roman rite, and the old one would be called the Extraordinary form.
The Pauline Mass - or Ordinary Form, as it is now properly called - does not have a distinction between High and Low Mass. You see, with the Extraordinary Form, there are very specific rules about what makes a High Mass and what makes a Low Mass. In the Low Mass, you could not have a deacon assisting, there was no music, no incense, etc. In a High Mass, you would have deacons, music, incense, and so forth. When they created the new Missal in the 60s, however, they did away with these strict requirements. They permitted the priest celebrating the Mass to choose whether to have music, or incense, whether a deacon would be assisting, etc., and he could mix and match. In other words, one could have a deacon but no music, or music but no incense. This isn’t possible with the Extraordinary Form, which has very specific instructions about these various things.
So basically, you have the Extraordinary Form, which has High (music, deacons, incense) and Low (no music, deacons, or incense) Mass, and you have the Ordinary Form, which can have any number of combinations of the different ceremonious aspects of it.
Here are a few of the differences between the Extraordinary Form and the Ordinary Form (there are more):
*]The EF MUST be said in Latin, the OF may be said in Latin or in the vernacular language of a place.
*]The EF REQUIRES the priest to face the altar, away from the people, while the OF permits him to face the people, which is how it is normally done in OF Masses.
*]The EF has several more prayers than the OF, which usually express the faith of the Church more thoroughly.
*]The OF is much more directly “interactive” than the EF. The OF has the congregation actively doing something most of the time, whereas in the EF the people will often be waiting in silence as the priest does something.
Now there are many different opinions that people have about these differences. Many prefer the EF because the Latin helps keep people in awe of God, and the priest facing away from (or, as it is properly said, with) the people better illustrates the fact that he is talking to God as the people’s representative, rather than merely talking to the people. Some prefer the OF because they are usually more active, whereas some prefer the EF because they believe that the atmosphere of silence is better. Some folks get very adament about this, but just remember that the Church permits either one. It is a matter of which Mass one discerns God calling him to in prayer.
Feel free to ask any additional questions.
Peace and God bless!