Let's preempt this all by stating that I am a Hispanist scholar with a focus on artillery/martial arts, literature and philology. Someone else might have a better answer.

Think of the trivium as your "soft" skills that help in critical thinking. The subjects were grammar (how we acquire knowledge), logic (how we process it) and rhetoric (how we discuss it). These were usually taught first, even in grammar schools for smaller kids, along with basic arithmetic and some Latin, especially prayers.

The university curriculum, aside from expanding on the trivium, also added the quadrivium, or "hard skills. These were arithmetic, music, geometry and astronomy. Even if the student arrived at the university already knowing some of these, the lectures ensured mastery before heading to the next level--philosophy.

So trivium and quadrivium were like undergraduate classes. People like Lope de Vega (he never finished) and Franisco de Quevedo, both great luminaries of baroque Spanish letters experienced this curriculum. Upperclassmen, like Antonio de Nebrija ("Gramática de la lengua castellana," 1492) and Juan Ginés de Sepúlveda ("Tratado sobre las justas causas de la guerra contra los indios," 1525-6) went on to do "graduate" work in philosophy, theology, canon law and medicine.

Together, trivium and quadrivium represent the 7 liberal arts. We call them other things now, even dividing them into more specific fields of research, but, as we learn more, we've only restructured human knowledge at the university level to better understand it.

Bibliography:

Ricardo da Costa. "Las definiciones de las siete artes liberales y mecánicas en la obra de Ramon Llull", Revista Anales del Seminario de Historia de la Filosofía. Madrid: Publicaciones Universidad Complutense de Madrid (UCM), vol. 23 (2006), p. 131-164 (ISSN 0211-2337)

Pierre Riché - Jacques Verger (2006) Des nains sur des épaules de géants. Maîtres et élèves au Moyen Âge.

Martianus Capella. Satyricon o De Nuptiis Philologiae et Mercurii et de septem Artibus liberalibus libri novem

There were seven liberal arts taught in Medieval Universities. These were divided into the Trivium, which included grammar, logic and rhetoric. These subjects formed the basis of the education and led into the Quadrivium. These were arithmetic, geometry, astronomy and music.

Grammar is the first of the seven liberal arts. It informs the student about the terms and usage of language. Once a solid understanding of this has been gained, logic teaches argumentation and deliberate thought. Rhetoric then teaches how to present arguments and to persuade. There are more abstract arts then the Quadrivium, and could be applied more generally.

I hope the separate aspects of the Quadrivium does not need explanation. Music was more the study of music theory, especially in relation to harmonics. Think Pythagoras rather than guitar lessons. Algebra was not introduced into Europe until after the Medieval period had waned.

The Trivium and Quadrivium were taught as a sort of undergraduate course. It was necessary for a student to master these arts, and have that mastery confirmed by the university, before they could enter further study, such as Law or Theology. I amn't totally confident, I don't have my source open or with me, but do believe that the Trivium was taught of as something analogous to a B.A. while mastering the Quadrivium granted someone a M.A..

This type of curriculum was nearly ubiquitous in Western Medieval Universities.

Source: Haskins, The Rise of Universities

Edit: A quick note. The curriculum grew out of Roman traditions, but can't really tell you much more than that. I studied this a few years ago, and it was more focused on the scholastics, such as Anselm, Peter Abelard or Aquinas, with only an introduction on the history of universities.

You might enjoy some of these related threads from FAQ ["Life at University"] (http://www.reddit.com/r/AskHistorians/wiki/dailylife#wiki_life_at_university).