Rosh Hashana is the beginning of the Jewish calendar year (there are three other dates that begin other aspects of the Jewish year).
Rosh Hashana is observed for two days, which are considered to be one continuous day. In Temple times, a cohen would offer a sacrifice but might not know until the following evening, after the new moon had been seen (or not), whether he had offered the Rosh Hashana offering.
Rosh Hashana focuses on the idea of God as King and of our relationship as Jews to that King.