This a good question. As you probably know, the history of Greece is a very long one. Hence, historians break up Greek history into various periods. With that said, it is important to keep in mind that not all historian agree with the dates of these periods.
Classical Greece for most historian starts with the rise of of the city states of Greece (poleis - plural). This is the time of the heights of Greek achievement. During this time we read of great statesmen like Pericles, philosophers like Plato and Socrates, and great building projects like the acropolis. During this time we also hear about great wars like the Peloponnesian War.
Hellenistic Greece comes later on. More specifically, it comes with the death of Alexander the Great in 323. After his death, his empire was broken down into various areas. For the most part, his generals took over. These were called the Hellenistic kingdoms.