Short answers, pulled from the story.
The Sanhedrin was a Jewish legislative and judicial assembly in the ancient land of Israel. It existed in two forms: a Great Sanhedrin of 71 judges that served as a supreme court, and lesser Sanhedrins of 23 judges appointed in each city.
During the Second Temple period, the Great Sanhedrin met in the Hall of Hewn Stones in Jerusalem, a building built into the northern wall of the Temple, half inside the sanctuary and half outside. It convened every day except festivals and the sabbath.
The Great Sanhedrin was disbanded in 425 CE when Gamaliel VI, its last president, died and Theodosius II outlawed the title of Nasi. The last universally binding decision of the Great Sanhedrin had been made in 358 CE, when the Hebrew calendar was established in fixed mathematical form.
During the presidency of Gamaliel IV (270-290 CE), the Sanhedrin dropped its name to avoid Roman persecution. Its authoritative decisions were then issued under the name Beth HaMidrash, meaning "house of learning."
Napoleon convened the Grand Sanhedrin in 1806 to give legal sanction to answers the Assembly of Notables had provided to twelve questions posed by his government. A proclamation issued on the 6th of October 1806 invited Jewish communities across Europe to send delegates. The body did not follow traditional halakhic procedures.
The Nasi, meaning "prince" and often translated as "president," was the chairman of the Great Sanhedrin. The office was created in 191 BCE after the Sanhedrin lost confidence in the High Priest. After the time of Hillel the Elder, the Nasi was almost invariably a descendant of Hillel and held authority over calendar regulation, appointment of judges, and representation before imperial authorities.