Western Philosophy began in 585 BC with the first philosopher: Thales of Miletus in Greece. From there it continued to spread throughout Greece. The great thinkers Plato and Aristotle created an entire system to explain all that existed in the world. Later on, Greek culture and ideas were spread by the conquests of Alexander the Great and adopted by the Roman empire. From the Roman empire, Christianity absorbed Greek philosophy as a way of explaining and defending its own ideas. It follows that we can divide Ancient Western Philosophy into four periods: Presocratic, Classical, Hellenistic and Roman.
Presocratic philosophy begins with Thales and continues until Socrates. It included such thinkers as the early atomists who believed that everything was made out of indivisible solids, sophists who believed that argument was pointless, a whole school that believed that change did not ever happen (nor could it) and another individual who believed that absolutely everything changed. Practically every possible opinion you could start with begin in this period. The Milesians were the first philosophical school. Other major schools include the Pythagoreans, the Eleatics, the Sophists and the Pluralists. Important individuals include Heraclitus, Parmenides and Phythagoras of Samos.
Classical philosophy begins with the questions of Socrates, and continues with the major figures of Plato and Aristotle. It also includes a number of minor thinkers who lived at the same time. Plato and Aristotle are the major figures of this time period. Socrates is only known by the writings of his students. Antisthenes was a student of Socrates, and Aristotle had a number of his own students as well. Since Aristotle was Plato’s student, and Plato was Socrates student they belong to a single project in philosophy. Their philosophy sought to create a complete philosophical system that would answer all of the questions of the presocratics while including all of their actual insights.
Hellenistic philosophy began after Aristotle and continued until the rise of the Roman empire. The two dominant philosophies of the time were Stoicism and Epicureanism. The philosophy of Skepticism had a strong influence in various academies. Other philosophies of the time included Platonism, Peripateticism (the philosophy of Aristotle), Cynicism and individual philosophers that do not fit into any particular school of the time. This time period focused on physics, logic and ethics. Most philosophers also included their religious beliefs as an integral part of their philosophy.
Roman (or Imperial) philosophy begins at about 31BC and continues until pagan philosophy is outlawed by the emperor in the east in 529AD. In the West, we like to include Augustine in the medieval period, but strictly speaking he belongs to the Roman period of Ancient philosophy. This period is noted for producing commentaries on previous philosophers so that their writings could be understood in the current period. It also has much in common with Hellenistic philosophy.