Esther 4:15 Then Esther bade them return Mordecai this answer,
16 Go, gather together all the Jews that are present in Shushan, and fast ye for me, and neither eat nor drink three days, night or day: I also and my maidens will fast likewise; and so will I go in unto the king, which is not according to the law: and if I perish, I perish.
17 So Mordecai went his way, and did according to all that Esther had commanded him.
Esther requested Mordecai to organize a three-day fast to prepare herself for the risky mission of going before the king without being summoned (which could result in a glad welcome ... or death).
The three days in quesiion were the 13th to the 15th of Nissan the first month for the feasts of the Lord
Esther 3:11 And the king said unto Haman, The silver is given to thee, the people also, to do with them as it seemeth good to thee.
12 Then were the king's scribes called on the thirteenth day of the first month, and there was written according to all that Haman had commanded unto the king's lieutenants, and to the governors that were over every province, and to the rulers of every people of every province according to the writing thereof, and to every people after their language; in the name of king Ahasuerus was it written, and sealed with the king's ring.
13 And the letters were sent by posts into all the king's provinces, to destroy, to kill, and to cause to perish, all Jews, both young and old, little children and women, in one day, even upon the thirteenth day of the twelfth month, which is the month Adar, and to take the spoil of them for a prey.
The 13th day of the 1st month Nissan, was when Mordecai learned of Haman's plot and how he engineered the King's decree against the Jews. Neither Haman nor the King knew at this time that the Queen was Jesish, as her cousin Mordecai who was her guardian, had instructed her to keep her Jewishness hidden.
Esther 2:20 Esther had not yet shewed her kindred nor her people; as Mordecai had charged her: for Esther did the commandment of Mordecai, like as when she was brought up with him.
The days of the fast therefore included 14th of Nissan, which, of course, was Passover, which is celebrated annually with a meal of lamb, with bitter herbs, etc. as prescribed in the Law of Moses.
Because Esther was keeping her Jewish identity a secret, she could not celebrate Passover. Therefore she fasted on Passover (and the day before and after).
However, in later years, the crisis having been marvellously removed by God's sovereign action, the 13th to the 15th of Adar were decreed to be days of feasting. Traditionally, however, the Jews practice a one-day fast on the 13th of Adar, the day before Purim which starts after sundown when the Jewish date changes to the 14th.
This is not without Biblical warrant:
Esther 9:31 To confirm these days of Purim in their times appointed, according as Mordecai the Jew and Esther the queen had enjoined them, and as they had decreed for themselves and for their seed, the matters of the fastings and their cry.
Apart from the three-day fast that Ester called for in Esther 4:16, there was another fast that the Jews used to observe, the pre-battle fast. There is Biblical warrant for pre-battle fasts.
The people of Israel used to fast whenever they were faced by war. The first such fast was when Moses fasted when he came to wage war against Amalek. The purpose of the fast was to affirm that the warriors of Israel do not prevail by physical or military strength, but only by lifting their eyes to heaven so that God Himself might give them the strength to defeat their enemies. This was the purpose of the fast by Israel at the time of Haman, when they gathered to defend themselves against those who were intent on their annihilation. The yearly pre-Purim Fast was fixed for generations to commemorate that first Purim pre-battle fast..
Whenever Israel is in danger from enemies who want to exterminate the Jews or to seize or divide their land, or their capital city, it is surely pleasing to God that we who know the centrality of Israel in the redemption plan of God, should fast to ask God for strength for Israel to be able to defeat their enemies.