Taanit Esther takes place one day before Purim, the 13th of Adar. The 13th of Adar was the day chosen by Haman to exterminate the Jewish people. So, we fast on that day to remember our anguish and to realize the big danger from which HaShem Almighty delivered us all.
There is more. When Queen Esther was informed of Haman's plan to eliminate the Jewish people, she decided to appeal to King Achashverosh, who did not fully know about Haman's plan. This desperate move was a suicidal mission, because anybody who would come near the King without being summoned by the King, could be legally executed due to security reasons. But there was virtually nothing else anyone else could have done to talk to the King and persuade him from stopping Haman to carry out the 'final solution'. Esther then decided to risk her life and approach the King.
Before Esther approached the King she asked every Jew to fast. Fasting--together with prayer--is what our Torah and Chakhamim instructed us to do in difficult circumstances. At the request of Esther, every Jew in the Empire fasted for 3 days and prayed for Esther's success. In remembrance of that event we call this fast day Ta'anit Esther.
Ta'anit Esther is technically a minor fast. Pregnant or nursing women do not do this fast. Also children, elders or anyone with any medical condition is exempted from this Ta'anit. Every other healthy adult should fast.
The fast of Esther starts tomorrow, Wednesday March 7th at 5.00AM and ends at 6.25PM .