Synagogues arose in the days of the Captivity in Babylon. The Temple of Solomon was destroyed, and the nation was scattered, there was a big need for places of instruction and worship wherever there were Jewish communities. After the Jews returned under the command of Cyrus of Persia the synagogues were continued both in Israel and in the Jewish centers in other lands. All larger towns had one or more synagogue. In Jerusalem, even though the Temple was there, there were many synagogues. They were presided over by a board of elders, or rulers. Early Christian meetings were modeled in part after the pattern of synagogues.
The word synagogue is actually a Greek word which means assembly or congregation. The origin of the synagogue is under speculation, some say it originated with Moses, others say was Ezra and Nehemiah. Still others say that it originated with the prophet Ezekiel in Babylon. However it originated there was a great need for the study of the Torah, organized prayer, and leaders who could answer the many asked questions concerning the Jewish people. During the Persian -period, and the Greek period, the synagogue became an important part of the Jewish community wherever they were. Even though the second Temple was in Jerusalem, there were synagogues spread throughout the cities and towns, under the control of the rabbis. When Jesus arrived on the scene he began his public ministry by coming into the synagogue at Nazareth and a reading from the scroll of the prophet Isaiah.
Wikipedia. Although synagogues existed a long time before the destruction of the 2nd Temple in 70 CE, communal worship in the time while the Temple still stood centered around the korbanot ("sacrificial offerings") brought by the kohanim ("priests") in the Holy Temple. The all-day Yom Kippur service, in fact, was an event in which the congregation both observed the movements of the kohen gadol ("the high priest") as he offered the day's sacrifices and prayed for his success. During the Babylonian captivity (586–537 BCE) the Men of the Great Assembly formalized and standardized the language of the Jewish prayers. Prior to that people prayed as they saw fit, with each individual praying in their own way, and there were no standard prayers that were recited. Rabbi Yohanan ben Zakkai, one of the leaders at the end of the Second Temple era, promulgated the idea of creating individual houses of worship in whatever locale Jews found themselves. This contributed to the continuity of the Jewish people by maintaining a unique identity and a portable way of worship despite the destruction of the Temple, according to many historians.