For what I am doing here, see this.
The Hellenistic period is a period consisting of the dominance of Greek culture, from roughly the time that Alexander the Great established his empire to roughly the time that the rising power of the Roman Republic becomes an Empire itself. As an arbitrary date, we can say it begins with the death of Alexander in India in 323 BC and ends somewhere between 146 BC, when Rome conquers the Greek homeland, and 30 BC, when Rome conquers the last major fragment of Alexander's empire, the Ptolemaic Kingdom of Egypt ruled by Cleopatra. From the philosophical perspective the period is dominated by schools, which begin as local institutions for education and gradually become major philosophical movements. Many schools had been founded by Socrates' students, but most of them did not last long; from that first generation only Plato's Academy survives. It is joined by three others, and these four schools of thought are what people usually mean when they speak of Hellenistic philosophy.
The New Academics: The purpose of Plato's Academy had never been simply to spread Plato's own ideas. It was instead a place for people interested in intellectual pursuits to come together and exchange ideas. Because of this, Plato's successors in the Academy often took it in new directions. The Platonic origin remained important, and the Academy preserved Plato's dialogues, but Plato's successors did not feel bound to follow him slavishly. The primary tone of the Academy in the Hellenistic period was set by Arcesilaus, who became the sixth head of the Academy somewhere in the middle of the third century BC. We don't have a precise idea of his views; he doesn't seem to have written anything, and descriptions of his teaching are somewhat confused. We do know that he advocated some form of skepticism; according to Cicero, he held that he knew nothing, not even whether he knew nothing, but if he did say that, it's not clear that it was meant to be taken completely literally. Whatever his precise views were, the Academy in this period becomes associated with a moderate skepticism -- the claim that we cannot know that things are true, but, at most, we can have reason to think them truth-like. Because this is a shift from the early days of the Academy, this phase of the history of the school is often called 'the New Academy', although this label is often reserved for the phase of the Academy that begins with the tenth head of the school, Carneades, who may be responsible for the moderateness of Academic skepticism; when this distinction is made, the phase begun by Arcesilaus is called 'the Second Academy' or 'the Middle Academy'.
The Peripatetics: The Peripatetic school was founded by Aristotle; they met at the Lyceum in Athens. The word 'peripatetic' means 'walking around'; apparently Aristotle liked to teach while walking. The Lyceum had a rockier history than the Academy; Aristotle's works were nearly lost several times, and, indeed, we are almost certainly missing a significant portion of his writings. One of Aristotle's major interests had been the natural world, and the Peripatetics of the Hellenistic period seem to have focused largely on this topic. A number of significant mathematicians and historians were associated with the school in one way or another.
The Stoics: Stoicism was founded by Zeno of Citium, and became one of the most thriving philosophical movements of the Hellenistic period. Called Stoics because they taught at the Stoa Poikile (which means 'The Painted Porch'), the Stoics aimed to become sages, completely wise individuals ruled wholly by reason and therefore always capable of rising above their passions, thinking with tranquil calm regardless of what happened to them. They argued vehemently against the Academics that it was possible to know things with certainty, and they argued vehemently against the Epicureans that pleasure was not the greatest good. They held that the universe was a living being -- God, in fact, and that divine Reason or logos pervaded everything. Our own reason is a sort of participation in this divine reason. Since they held that Nature is God, and thus full of divine Reason, they often describe the good life as living according to Nature or living according to Reason. Because of their interest in reason, the Stoics made a number of advances in logic; however, we only have fragments of their work in this field.
The Epicureans: Epicurus of Samos argued that the good life consisted in a life of tranquil pleasures. Good and evil are explained in terms of pleasure and pain. The Epicureans were also materialists: they held that everything consisted of atomoi, indivisible particles, moving in a void. They were the most anti-Socratic of the Hellenistic schools, but Epicurus may have been influenced by the school of Aristippus, one of Socrates' students; Aristippus also held that the good life consists in pleasure, although he seems to have allowed a wider range of acceptable pleasures than Epicurus. He certainly was influenced by the thought of Democritus, an earlier philosopher who had also proposed an atomic theory of the universe. Because Epicurus' students in Athens met in his garden, the Epicurean school was often called The Garden. Interestingly, the Epicureans were the most static school of the Hellenistic period: their doctrines changed very little over their entire history. Part of this seems to be because students in The Garden were required to swear an oath to uphold Epicurus' basic teachings. A good description of Epicurean doctrine is found in Epicurus' Letter to Menoeceus.
In addition to these four major schools, there were two other significant philosophical movements; both of them were less organized than the above schools.
The Cynics: The first full Cynic is usually said to be Diogenes of Sinope. He seems to have been heavily influenced by Antisthenes, one of Socrates' students, and to have argued that happiness consists in living according to nature, which they understood as complete self-sufficiency, free of all social convention and the cares that come from having material possessions. The name 'Cynic' literally means 'dog-like' in Greek; we don't know for certain why they were called this. Some say that it was because Antisthenes taught at the Cynosarges gymnasium; but it may have been an insult (that they lived like dogs), which they began to wear with pride. Zeno of Citium was almost certainly heavily influenced by Cynic views.
There are many anecdotes and legends about Diogenes of Sinope, who was once called "Socrates gone mad". In statues and paintings he is usually depicted holding a lantern; this stems from the most famous of these anecdotes, in which Diogenes goes around in broad daylight with a lit lantern, looking into various corners of the city. When asked what he is doing, he replies with something along the lines of, "I am looking for an honest man; I still have not found one."
The Skeptics: One of the philosophers who travelled with Alexander the Great was Pyrrho of Elis. We don't know precisely what his views were, but it is said that he advocated complete suspension of judgment: dogmatic belief was a source of misery, because human beings cannot know anything, so happiness was to be found by merely going with the appearances and not being dogmatic about anything. This inspired a philosophical movement in the late Hellenistic period, under a philosopher named Aenesidemus, which was called Pyrrhonism. We don't have a clear history of how the later movement was related to Pyrrho himself. It seems plausible, however, that in between the two there were scattered skeptics not associated with the particular version of skepticism found in the New Academy.
The actual physical institution of the Academy seems to have been destroyed due to war in 86 BC (as was the Lyceum), although teaching went on elsewhere. About this time Antiochus of Ascalon broke away from the skepticism of the New Academy in order to revive the teaching of the Academy under Plato: what he founded is sometimes called the Old Academy because of this. What he actually started was a philosophical movement that came to be called Middle Platonism; it is this movement that will come to dominate the early Roman Imperial period.
For times were changing. The Academy and the Lyceum had been ravaged in the First Mithridatic War, a failed rebellion of Greek cities against Rome's ever-increasing power. And after the death of Cleopatra VII of Egypt in 30 BC, Rome ruled the Mediterranean Sea with complete sway.