The Jewish calendar actually has more than one "new year", depending on what you're counting.
1 Nissan is the new year for months; when the torah says "in the seventh month" (or some other numeric counting) it's counting from this. Nissan is the month of our redemption from Egypt, a defining moment in our history. The torah explicitly refers to the month of the exodus as the "beginning of the months for you".
1 Tishrei is the new year for the world. The world was created on this date (or, some say, five days earlier and man was created on this date). Sabbatical (shmita) years are counted from this date.
15 Shevat is the new year for trees. This is the date we count from for purposes of tithing fruit. Also, during its first three years a tree's fruit is forbidden (this is called orlah), so it's important to know when to count from. We don't track the planting of individual trees; they all count from the same date. (I've just learned that some count orlah from 1 Tishrei instead.)
The mishna (Rosh Hashana 1:1) lists 1 Elul as the new year for tithing cattle, but this opinion is later rejected (h/t DoubleAA).
For more information, see My Jewish Learning.