I’m no expert on the exact details, I’m sure you will get better responses from others here before too long… but since it looks like you are looking for mostly the “external” differences–the primary differences that you would notice as a typical lay Mass-goer, I will try to give you a brief overview of the basics.
First, very short history lesson. The “Tridentine” Mass dates officially to the Council of Trent back in the 1500s, but the rite itself has elements that can be traced all the way back as far as the Church Fathers (including Justin Martyr back in around 120AD). So, the “Tridentine” Mass is historically speaking the traditional rite of the Catholic Church.
Enter Vatican II… This council, which is to be considered an authentic Church council (contrary to what many traditionalists say), as part of its work attempted to initiate reform of the liturgy. Sadly, some of this reform has been misinterpreted by liturgists (and worse than that…) but the correct teaching of Vatican II does include a “New Order” of the Mass. This “Novus Ordo” Mass is now in use throughout most of the United States (except for a few parishes with permission to continue to celebrate the Tridentine Mass, and a few parishes who DO NOT have permission to celebrate the Tridentine Mass but do anyway)
So what are the big differences (at least the obvious ones to all the laity who attend)??
First, the language. The Tridentine Mass is ALWAYS celebrated only in Latin, with only the homily (and the Gospel?) in the vernacular (in America, English language). Today, most places that have the Tridentine Mass offer the use of a Missal that has the translations from Latin to English, so you can sort of follow along. However, you must remember that under the Tridentine rite, “following along” is not of the same importance as it is to many who go to the current Novus Ordo Mass… you do not need to truly follow every line in Latin with understanding, nor are you supposed to.
Second, the length. The rite of the Tridentine Mass is more elaborate and complex than the current Novus Ordo Mass–and thus lasts longer when celebrated correctly (the one in my area is usually between 1 1/2 - 2 hours long).
Third, the vestments. The Tridentine clothing for the celebrating priests and servers is the “old” style, very interesting looking and eye-catching. Also, there can be no female altar servers under the old rite–and there are a lot more servers than we typically see in the new one. The parish in my area tends to have between 10 and 15 altar boys at each Mass.
Fourth, the position of the priest. The celebrant of the Tridentine rite generally faces the altar/tabernacle (usually towards the east), not towards the people.
Fifth, the congregation ONLY receives Holy Communion on the tongue (not in the hand), and only while kneeling at the communion rail. There is no “pew-by-pew line-up” like at the Novus Ordo Mass, instead generally people just go up and wait in line as they feel ready to receive. This may sound inefficient, but in actuality it moves just as quickly (I think) as the modern style. Plus, a bonus of this method is that it relieves the pressure of someone feeling as though they “have” to go to Communino even if they are in a state of mortal sin (the “everyone in my pew is going… people will notice if I don’t” problem).
Sixth, generally speaking, nobody attending a Tridentine Mass would think of wearing jeans, t-shirts, shorts or any other casual clothes. Many women who attend wear a hair veil (though no, it is not required or necessary–I do not wear one myself). Large families (with 8 or more kids!) are usually in attendance.
This is NOT to say that the Novus Ordo Mass is “bad” or “wrong” or even “invalid” (!) as some have said. The Novus Ordo Mass done as it should be according to the rubrics is a beautfiul and reverent celebration of the Sacrifice of Calvary and there are many many parishes that do this to the best of their ability.
However, the current Pope has made it clear that the tradition of the Tridentine Mass rite is to be an option for those faithful who would prefer to worship God in this manner. Unfortunately, many clergy do not see the need to fulfill this desire of the Holy Father, and refuse to allow an official diocesan-approved Tridentine Mass to be offered in their parish or diocese. It is most often in these cases and dioceses that the extreme traditionalists (who have split from Rome, saying that the Novus Ordo Mass is invalid and that there is no authentic Pope in the See of Peter) have large followings.
For those in the Archdiocese of St. Paul/Minneapolis, our indult (approved) Tridentine Mass is held every Sunday morning at 11:30am at St. Augustine’s parish in South St. Paul. Fr. Echert (of EWTN fame ) and Fr. Ubel are the typical celebrants, both of them are very good and holy priests.