In the 13th century, it became somewhat normal to try to write a comprehensive summary of the theology of the time, debating the questions of the day. Thus, figures like the Franciscan Alexander of Hales (1158-1245), Albert the Great (1200-1280), Henry of Ghent (1217-1293), and Thomas Aquinas (1225-1274) all wrote some form of summary of theological questions—though their forms do differ. 

The most famous of these throughout later history (for various reasons, some accidental, some related to the clarity and insightfulness of the text) was the Summa theologiae of Thomas Aquinas. This work is a massive outline of the science of theology, considering questions as varied as the nature of the Trinity to the natural law, to the various virtues, to the incarnation of Christ, to the sacraments. With the passage of time, it would become very influential in Catholic theology. (Although, it was not the only story in Catholic theology, for the Franciscans, Augustinians, and others all had their own schools of thought as well. Still, Thomas was the the thinker who won ecclesiastical approval most vociferously.)

The great Dutch humanist Desiderius Erasmus is perhaps most well known for his The Praise of Folly, a satire of the somewhat superstitious practices of Catholics in his day as well as the practices of scholars in that time. He also took part in a lengthy disputation with Martin Luther regarding the freedom of the human will in light of human sin. He also was extensively involved in working with Greek and Latin editions of classical, patristic, and biblical texts. While Erasmus was a critic of aspects of the Christian world of his day, he remained a Catholic all of his life. He was frustrated with corruption and wished to reform the Catholic church without the kind of radical measures undertaken by Luther. He was a friend of Sir Thomas More.

For some questions in the humanities, you just need to apply your general knowledge to help eliminate people. The Summulae Logicales of Peter of Spain were very important for many centuries—even though it is not completely clear just who the author, in fact, was! In any case, you may not know of Peter, but you should know that Kant is an 18th / 19th century figure, that Seneca is a Roman of the first century AD, and that Aristotle is a Greek from the fourth century BC. This leaves you with two options. Now, Aquinas is not known for writing texts in logic (although there are some logic texts falsely attributed to him, and he did write commentaries on several of Aristotle's logical works). Hence, if you at least know in general that Aquinas is not known for being a famed logician, you can guess that Peter of Spain is the correct answer—and you would be correct!

William of Ockham (1825-1347) is best known in the popular mind for the so-called "Ockham's razor," a common sense idea that we shouldn't just suppose that there are many kinds of beings in the world necessary for explaining our theories. He is, however, also known (for related reasons) as a "nominalist," that is, as someone who thinks that universal notions like "dog" and "tree" do not subsist in reality but only arise in the mind, which compares individual things that are maximally similar—hence, calling those individual things by a given universal name.

He also wrote the highly influential Summa logicae, which outlined a form of logic that was congenial to this outlook (as well as other aspects of his theory of mind and metaphysics). If you ever doubt that there was sophisticated philosophy in the middle ages, crack open Ockham's works, and you will see that he (along with many others) were quite agile minds indeed!

The great Dutch humanist, Desiderius Erasmus, engaged in a long series of polemics against the Protestant reformer, Martin Luther, on the subject of the freedom of the human will. Because of his particular theological commitments, Luther believed that the human will was so perverted because of the Fall (i.e. the supposed first sin of Adam and Eve) that it was bound by sin and Satan, requiring redemption if it were to be free again. Luther's great work in this debate is known as On the Bondage of the Will or On the Captivity of the Will.

The theologian Thomas Aquinas wrote much in his brief forty-nine years of life. Many people are introduced to him through a brief passage in his Summa theologiae in which he proposes five possible "ways" for proving God's existence. In various manners, these five proofs are based on ways that someone can start with human experience and prove from that finite, changing experience how there must be an unchanging God. 

Many of the other thinkers listed in this question as potential answers had interest in matters similar to this as well. Most directly pertinent for this matter is Anselm of Canterbury. Anselm is well known for his so-called "ontological argument" for God's existence (though the title is a bit of a later attribution). His general idea was that so long as you can have an idea of, "Something than which nothing greater can be thought," you can prove that such a great thing must indeed exist—precisely because it is so great that it is perfect and hence has the perfection of existence.

Interestingly, many of the works of Aristotle returned to the Latin West by means of Arabic translations. Often the Latin editions were Latin translations of Arabic translations of Syriac translations of the Greek. Eventually, better Greek editions came to the West, but the Islamic commentators on Aristotle remained very important.

The two greatest figures (though there were others) were Ibn Rushd (or Averroes) and Ibn Sina (or Avicenna). Many of the works of Aristotle were very difficult to read. Part of this was due to the difficulty of the texts but part of it was also due to the poor translations (which were often very literalistic). Ibn Rushd had written a number of commentaries on Aristotle's works. Latin translations of these commentaries were often consulted by medieval philosophers so as to help in the parsing of Aristotle's texts. (Indeed, in some editions of Aristotle's works, Ibn Rushd's commentaries were included in the margins.)

Ibn Sina was a unique and creative Persian philosopher who combined Aristotle's thought with many other forms of philosophical thought (especially those of Neo-Platonism). Ibn Rushd thought that Ibn Sina's thought was not faithful to Aristotle. Nevertheless, Ibn Sina's writings on natural philosophy and metaphysics (or "first philosophy") were very important for medieval thinkers who were learning anew the details of Aristotle's thought.

The Middle Ages was a variegated and differentiated period, filled with many thinkers of quite differing temperaments and ideas. The general theme that united the period, at least in its time of greatest development, was the advent of the universities and the eventual division of camps of thinkers into various "schools"—i.e. schools of thought such as Thomism, Nominalism, Scotism, Albertism, and so forth. For this reason, the period is often identified with the title "Scholasticism," though you should always remember that this one term really applies to many different perspectives and outlooks.

By the 13th and 14th century, the liberal arts curriculum had generally stabilized and arrived at what became its classical formulation. These topics were studied by all before going on to higher studies such as theology, law, or medicine. Also, during this same period, other topics were added to the arts, especially natural philosophy but also a study of other works of Aristotle such as the Metaphysics and the Nicomachean Ethics.

The liberal arts were split up into two main branches. The first consisted of three subjects pertaining to logical expression, while the second consisted of four subjects pertaining to mathematical matters. The first three were called the "trivium", literally meaning "the three subjects": grammar, logic, and rhetoric. The latter were called the "quadrivium", literally meaning "the four subjects": arithmetic, geometry, astronomy, and music (understood as a science of mathematical proportions).

The Sentences of Peter Lombard (1096-1160) was the single most influential textbook for many, many centuries. It was a compilation of sources, systematized for the study of theology. In the universities, it became the custom that the "dissertation"—it really was not that, strictly speaking—required the writing of a commentary on all four books of Lombard's Sentences. These works were often massive and involve a great deal of philosophical and theological speculation. Often these commentaries involved thousands upon thousands of pages. This was a requirement of university students for centuries—even Martin Luther wrote a commentary on this work, as did other students of his time!