Socrates and Plato (~400 B.C.)
Historical Context: Plato was born cica 427 B.C. at the end of what is often called the Golden Age of Athens. This age ended with the defeat of a democratic and industrial Athens against a militaristic and agricultural Sparta in the Peloponnesian War. The defeat led to a brief reign of terror by a group of aristocrats who were ultimately overthrown and democracy was re-established. The newly democratic Athens sentenced Socrates to death because of his anti-democratic views. Following this, Plato too fled from Athens and after trying unsuccessfully to implement his idea of the philosopher-king in the real-world, came back to Athens to take to full-time teaching.
Philosophy: I will mostly talk about Plato's philosophy as he was highly influenced by Socrates and hence his philosophy mostly includes that of Socrates. The heart of Plato's philosophy is that Virtue is Knowledge.
Descarters (~17th AD)
Historical Context: After the time of Plato, the Greeks were conquered by the Macedonions, who were in turn defeated by the Romans. The decline of the Roman empire coincided with increasing dominance of Christian beliefs. The church destroyed many Greek and Roman writings, charging them with being pagan and un-Christian. For over a thousand years, from the 4th century to the 14th century, the Church controlled the social and cultural life of Europe. Plato's and Aristotle's writings re-emerged in the Western world via the Muslims. Saint Augustine and Saint Thomas led the construction of a new philosophy combining Catholic beliefs with the re-discovery of Plato and Aristotle's philosophies. All this led to what is now called the Renaissance.
Descartes was born in this fast changing world. The centuries-old dominance of the Church on the Western life was on a decline. A new empirical science based on mathematics and deductive reasoning was being built by people like Copernicus and Galileo. However, the Church was still quite powerful and Descartes deemed it too risky to offend the Church.
Philosophy: Descartes wanted to explain the world rationally. Starting from self-evident truths, he wanted to mathematically founded the field of mathematical physics.
Hume (~1740 AD)
Historical Context: The period of roughly a hundred years starting from Descartes' death and ending with Hume's death was a period of utmost optimism and self-confidence in western philosophy. So much so that this period has also been called the Age of Enlightenment (in the West). This optimism was rooted in fundamental discoveries in science, rise of a self-made middle class and wildly new opportunities for trade and commerce. All these events contributed to a new found confidence in the superiority of human reason. Newton was the poster child of this belief as he had succeeded in demonstrating that every mechanical interaction, whether on earth of between celestial bodies, could be explained by a small set of simple laws.
Philosophy: Hume was going to puncture massive holes in this belief in the superiority of rationality and the human mind. Hume was the greatest of a line of British empiricists, a line that included the likes of John Locke and George Berkeley. The empiricists poses a seemingly innocent question, "How do you know?". In the hands of Hume, this question turned out to be a wrecking ball that devastated philosophic optimism of the time.