תִּשְׁרֵי (transliterated Tishrei or Tishri) is the 7th month of the Hebrew year, is 30 days long, and corresponds to September or October on the Gregorian calendar.
The holiday that occurs on the 1st day of Tishrei is called Rosh Hashana, the Jewish New Year. Rosh Hashanah is a major holiday.
The first day of Tishrei is not considered Rosh Chodesh. Rosh Chodesh is a minor holiday that occurs at the beginning of every month on the Hebrew calendar. In the case of Tishrei, the major holiday of Rosh Hashana takes precedence.
Later in the month we observe Tzom Gedaliah on the 3rd, Yom Kippur on the 10th, Sukkot (from the 15th-21st), and Shmini Atzeret (on the 22nd) and Simchat Torah (on the 23rd in the Diaspora).
For more info, read The Month of Tishri – Judaism 101.