The Hellenistic period is what scholars refer to as the time between Alexander the Great and the Roman Empire. Spanning some 300 years, the Hellenistic period was marked by a number of large kingdoms, all heavily influenced by classical Greece, which were established by Alexander's generals after his death in 323 BCE.
Most notable of the Hellenistic kingdoms was Egypt, seized by Alexander's friend and general Ptolemy I, who established the Ptolemaic dynasty in Egypt and founded the Great Library of Alexandria. The last of the Ptolemies was Cleopatra VII, whose suicide in 30 BCE is generally seen as marking the end of the Hellenistic era.
The Hellenistic era left great influences on Roman culture, particularly in the eastern Empire. Likewise, a large part of Jewish history was spent under control in the Seleucid Empire; the political realities of Seleucid (and later Roman) domination formed much of the background behind the time of Christ.