The 1^st of Nisan marks the beginning of a year for calculating the reign of a Jewish king. If a Jewish king began to reign in say Teves then by the time Nisan comes around he enters his second year of reign even though he has only been ruling for three months. The 1^st of Nisan is also the start of a new year for calculating the Yamim Tovim. When the Torah talks about the seventh month it refers to Tishrei.

The 1^st of Elul marks a new year for calculating maaser beheimah – tithing sheep and cattle.

The 1^st of Tishrei is the start of a new year for people – the day when Hashem judges us and decides what our lives will be like the coming year. It is also the starting point for calculating the years of shmittah and yovel as well as maaser of wheat and vegetables.

The 15^th of Shvat is the beginning of a new year for trees in regard to tithing its fruits.

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