Second Temple Judaism is Judaism between the construction of the Second Temple in Jerusalem, c. 515 BCE, and its destruction by the Romans in 70 CE. The development of the Hebrew Bible canon, the synagogue, Jewish apocalyptic expectations for the future, and Christianity, can all be traced to the Second Temple period.
(Note: dates and periods are in many cases approximate and/or conventional)
The period of the First Temple ended in 586 BCE when the Babylonian king Nebuchadnezzar captured Jerusalem, destroyed the Temple of Solomon, and deported the elite of the population to Babylon (the "Babylonian exile"). In 539 BCE Babylon itself fell to the Persian conqueror Cyrus, and in 538 BCE the exiles were permitted to return to Yehud medinata, as the Persian province of Judah was known. The Temple is commonly said to have been rebuilt in the period 520-515 BCE, but it seems probable that this is an artificial date chosen so that 70 years could be said to have passed between the destruction and the rebuilding, fulfilling a prophecy of Jeremiah.
The end of the Persian period is conventionally dated from Alexander the Great's conquest of the Mediterranean coast in 333/332 BCE. His empire disintegrated after his death, and Judea, including Jerusalem, fell to the Ptolemies, the descendants of one of Alexander's generals who ruled Egypt. In 200 BCE Israel and Yehud were captured by the Seleucids, the descendants of another Greek general ruling Syria. Around 167 BCE, for reasons that remain obscure, the Seleucid king Antiochus IV Epiphanes attempted to suppress Jewish worship; this provoked a Jewish revolt (the Maccabean Revolt) that eventually led to the effective end of Greek control over Jerusalem.