The month of Shevat (חדש שבט) is the eleventh month of the Jewish calendar (counting from the month of Nisan). The name "Shevat" (שְׁבָט) was adopted by the Jewish people sometime during the Babylonian exile, though it is explicitly mentioned in the Scriptures:
On the twenty-fourth day of the eleventh month (בְּעַשְׁתֵּי־עָשָׂר חדֶשׁ), which is the month of Shevat (חדֶשׁ שְׁבָט), in the second year of Darius, the word of the LORD came to the prophet Zechariah... (Zech. 1:7)
The month of Shevat is considered important since the Scriptures state that Moses began his summary of the Torah (i.e., Mishneh Torah, or the sermon recorded in Book of Deuteronomy) on the first day of this month:
"These are the words that Moses spoke to all Israel beyond the Jordan in the wilderness ... in the fortieth year, on the first day of the eleventh month [i.e., Shevat] (בְּעַשְׁתֵּי־עָשָׂר חדֶשׁ)" (Deut. 1:1,3)
Because of this, the sages have long associated the 1st of Shevat with the holiday of Shavuot (i.e., the sixth of Sivan), since on both these dates God appealed to Israel to receive the message of the Torah. Moreover, since Jewish tradition says that Moses preached the contents of Deuteronomy for 37 days, the month of Shevat (which lasts 30 days) until Adar 7 is considered an opportune time to renew your study of the Torah.