The Torah tells us that the month of Nissan is the first month in the Jewish calendar. The Mishna tell us (in the beginning of Rosh HaShana) that common practice was to incorporate into all documents the date in reference to the number of years since the local government had been founded. Each time a king died, and a new ruler was appointed, legal documents would reflect the number of years since the new king had been crowned. If a new government was established on July 4th, for example, one would not wait until the following July 4th to reflect the new ruler. If there was a Jewish king ruling over Eretz Yisroel, Rosh Chodesh Nissan would be considered the start of the new year. Regarding all other governments, the New Year began with Rosh Chodesh Tishrei. The Ran, commenting on the Rif (beginning of maseches Rosh Hashana) understands that this was not merely the conventional practice, rather it reflected a halacha.
The Netziv, in his commentary on the Torah, explains the idea behind this halacha as follows: When God created the entire world in six days, He instituted all the rules of nature. Included among the laws of nature are the principles of mathematics, physics, chemistry, and even the rules of history. Just as a chemistry expert can predict accurately the reaction that will follow the mixing of two chemicals, so too, one who understands well the rules of history can predict particular events, even the rise and fall of a particular nation. Years prior to the Japanese attack on Pearl Harbor, an American politician predicted, in writing, that early on a Sunday morning in December, in so many years, the Japanese will attack the American army base. He explained logically why it would take place on a Sunday morning, and why in December, etc., and his prediction was accurate. This is what the Rabbis of the Talmud meant when they wrote that all the nations of the world are "under the control of the stars and the constellations." The history of the nations is controlled by the natural rules of history, instituted by the Creator at the time of creation. It is for this reason that when recording the years of the non-Jewish kings, (i.e., non-Jewish history), the Rosh HaShana is the first day of Tishrei. In other words, the day on which we celebrate the completion of the creation, and the day on which the rules of history began.
However, the history of the Jewish people is above nature. The month of Nissan represents nissim (miracles), and we were commanded to count the months starting from Nissan (as the first month). Ramban, in his commentary on the Torah, writes that in his opinion this point is most crucial to the hashkafa of our religion; namely, that Jewish history is "lemala min hateva." Does it make sense that if the Jews violate the laws of shemittah they lose their control over Eretz Yisroel? Is it logical that if a Jew observes shabbos and kashrus his parnassa will be taken care of? The answer is that "ein mazal leyisroel," that the natural rules of history do not apply to the Jewish people.
Jewish history starts its New Year with Rosh Chodesh Nissan, the month of miracles, to emphasize the idea that all of Jewish history is made up of miracles. Our history begins and ends with the month of Nissan, the month of nissim.