Located in his work The Republic, the allegory of the cave is one of the most famous ideas associated with the Greek philosopher Plato. Taking place in a dialogue with Socrates, the allegory describes people chained inside a cave, forced to watch only the images projected by shadows on the cave wall and assuming this is reality. Plato has Socrates assert that a philosopher's role is to bring people out of the cave and into reality by destroying what they assumed by only watching the shadows in the cave.

During the period of the Pre-Socratics (literally, the philosophers before Socrates), many Greek thinkers focused on trying to figure out the nature of the world. They wanted to figure out how one thing (like a human person or even a rock) could be made up of many parts (like the organs, tissues, or even mere molecules involved in the former). When you read the Pre-Socratics, they can seem pretty crazy, but they were on to something, so to speak. They wanted to figure out just what it means for one thing to be many and for a thing to change (but still remain this or that particular thing).

Heraclitus was well known for cryptic sayings. Indeed, he was so well known that a minor American philosopher who used to live in Washington, DC was called the "Heraclitus of the Potomac," meaning that he was as difficult to understand as was Heraclitus.

In any case, Heraclitus is well known for some form of the saying, "You cannot step into the same river twice." One of Heraclitus's insights (among many others) was that change plays an important role in everything that we experience. Thus, when you walk into a river, it has certain molecules. When you step in again, those molecules have gone by and new ones have replaced it. Of course, if we carried this saying to the extreme, we would also say that you and I are quite different people at every moment of our lives—for our molecules change quite a bit.

The great systematizer, Aristotle, would see how this is true—but only in a certain respect. He would do this by analyzing how there are different kinds of causes involved in everything—one of which is the matter (which does indeed change).

In the Apology, the trial of Socrates is recounted as he defends himself against his accusers. The course of his speech, he refers to himself as a "gadfly" sent to wake up Athens. The image of the gadfly is used in Greek mythology, but the point in Socrates's use of the expression is that he was sent (by his supposed "spirit guide" who goaded him into practicing philosophy) to question the Athenians and unseat them of their pride and supposed knowledge. He was a kind of "annoyance" meant to pull them down from their conceits.

The Consolation of Philosophy was written by Anicius Manlius Severinus Boethius during his period of imprisonment at the hand of his former "employer", Theodoric the Great. In the text, Boethius is visited by Lady Philosophy, with whom he discusses a variety of Philosophical topics. The work is interspersed with related poetry as well. Throughout their discussions, Boethius is consoled that true felicity is not found in the passing and capricious happenings of this world (and the opinions of those in it).

In Greek, the word Organon means "instrument." The followers of Aristotle came to call his logical writings the Organon because logic is an instrument for all of the other sciences. This canon of logical writings would become a standard canon for teaching logic in the Middle Ages after Aristotle's works were rediscovered. (Only parts of his works survived into the early Middle Ages.) There were other forms of logic that developed after Aristotle, especially that of the Stoics. Likewise, in the Middle Ages, there was the development of numerous logical topics not directly covered by Aristotle. In addition, contemporary logic tends to ignore much of Aristotle, though even it is not without reliance on the old Greek philosopher. 

Interestingly, Aristotle himself thought that he was a pioneer in logic and that he was breaking new ground. Indeed, he says as much in a charming way in his Sophistical Refutations—a treatise dealing with sophistries (i.e. arguments that appear to be valid when they are not).

After the death of Plato, his Academy lived on, and there were many revivals of Platonism—and, indeed, there will always be many revivals of Platonism, at least if one notices the patterns in history. Among these revivals, there was the neo-Platonic school of thought, of which Plotinus is the most well known advocate. Often neo-Platonists believed that they were explaining just how Aristotle's own work is related to that of Plato (though clearly giving pride of place to Plato). In his Enneads, Plotinus expresses a broad cosmology, drawing all of experience (often using Platonic, Aristotelian, and Stoic notions) into relation to the "One" from which all reality emanates.

Plotinus was well known for his kindness, and the Enneads is a testimony to his contemplative grandeur. Through direct and (mostly) indirect paths, Plotinus would come to influence all of later Western philosophy. The most important conduits for neo-Platonic thought (though there were others) into the Middle Ages were Augustine of Hippo and the unknown maybe-Syrian-monk Pseudo-Dionysius the Areopagite. These two thinkers would be authorities to the Middle Ages, transmitting much neo-Platonism. Pseudo-Dionysius the Areopagite was particularly deep in his neo-Platonism, though his influence paled in comparison to that of Augustine, who was an uncontested father of early Christianity in the Latin West.

As a short side-note, Pseudo-Dionysius the Areopagite is so-called because of his claim to be a certain Dionysius the Areopagite who was converted by St. Paul. Later thinkers came to realize that he could not be such a person. He clearly drew on the work of Proclus, an exponent of neo-Platonism and could not have lived at the time of Paul of Tarsus. Thus, one pre-appends "pseudo" to his name.

The term "metaphysics" literally means "after physics." It is attributed to a work of Aristotle which, properly speaking, does not have that name given to it by its author. It deals with the problem of "being as such." For this question, we do not need to understand the details of this difficult treatise. Instead, it is enough for us to know something about the history (or at least theories concerning the history). The Metaphysics received its name really for two reasons.

On the one hand, it is the text that is to be studied after Aristotle's Physics. The Physics studies changeable being (or "mobile being", as later medieval writers interpreted Aristotle). The Metaphysics goes beyond this—it studies "being as such" or "being insofar as it is being [i.e. not merely mobile].

On the other hand, the name Metaphysics may well be derived from the fact that one of Aristotle's later followers, Andronicus of Rhodes, happened to organize Aristotle's works such that the text called the Metaphysics came after the Physics.

Really, both explanations can go together. Only one option has been provided that explains this phenomenon. (Be careful not to pick the option that says that the Metaphysics was named for its place within the various physical sciences!)

Plato wrote many dialogues on a great number of philosophical topics. His philosophical genius was matched only by his great stylistic ability to craft dialogues among various interlocutors. The Republic is his great work on political philosophy, one that is read by many people and would be a foundation text (whether accepted or rejected) for much of European political thought through the ages. This lengthy text deals with problems of justice, the way that various classes in a city should interact, how the city reflects the inner life of the human soul, and many other topics. Like much of Plato, the work is very airy and lacks the pragmatism that one finds in his student Aristotle's work The Politics. Nevertheless, the Republic remains an important work of political thought, working through numerous topics pertinent to the civic order—always with the deft expression that one expects from the able pen of Plato.

Sextus Empiricus actually presents us with a quite developed account of what was known as Pyrrhonian Skepticism. The word "skeptic" has a kind of negative connotation, as though the skeptical person really doesn't care about the truth. This is not really what it meant in Greek thought, however. The skeptic doubted that we could come to know anything absolutely. Sextus held that one could always settle for either side of an argument. He defended this position by presenting various "modes" of arguing that show this fact. 

The skeptic was really someone who "kept looking." The Greek roots for the word indeed reflect this outlook, for it means "to think" or "to consider." The Skeptic is someone who does not stop thinking or considering a given matter.

If you pick up any Latin philosophical or theological work from the mid-13th century onward (all the way through early modernity), you will find many references to philosophus or "The Philosopher." This refers to Aristotle, whose philosophical works in the Middle Ages were of utter importance to the development of much of medieval thought. Although many medieval thinkers combined Aristotle's thought with that of others, his corpus was the sine qua non of education during this period of history. It shaped much of Western thought, and it cannot be overestimated how much it shaped the curriculum of learning all the way through the time of early modernity.