The minhag yisrael is to fast on the 13th of Adar in order to commemorate the fast that the Jews fasted before going to war in the days of Mordechai and Ester. It is a common misconception that this fast was accepted by the Jews for all future generations during the time of Esther, as it is stated in the Book of Esther: They had established for themselves and their descendants the matters of the fasts and their cry (Ester 9:31). According to most rishonim, this verse actually refers to the four fasts which relate to mourning for the Temple. Rather, this fast is a Minhag to commemorate the fast that the Jews fasted before going to war in the days of Mordechai and Ester.  If Purim falls out on Sunday, the fast is moved up to Thursday. 
- It’s the Minhag of Yisrael to fast on the 13th of Adar in commemoration of the fast of the Jews in the time of Ester and Mordechai and it’s called Tanit Ester. 
- Normally, one may not make a wedding on the night before a fast day, but one may make a wedding the night before Taanit Ester. 
- Although some say that one shouldn't eat more than usual on the night before the fast, nowadays there is also room to be lenient. 
- Although some are strict to observe all of the stringencies of Tishba BeAv on the other fast days, these stringencies don’t apply to Tanit Ester. 
- Tachanun and Avinu Malkanu are not recited at Mincha on Taanit Esther when it is the day before Purim. 
If Purim Falls out on Sunday
- If the fast falls out on Shabbat, meaning Purim falls out on Sunday, it’s pushed to the Thursday before Purim, not to the Friday.  Someone who mistakenly ate on Thursday should make up the fast the next day, Friday. 
- Someone was traveling and didn’t know Taanit Ester was pushed off to Thursday should make it up by fasting on Friday. 
Those exempt from Tanit Ester
- Pregnant women (after 3 months of pregnancy, or within the first three months if the woman is experiencing pain from aches or vomiting) and nursing women (24 months after birth, even if she stopped nursing or there was a miscarriage) are exempt from fasting on Tanit Ester. 
- Children under Bar/Bat Mitzvah aren’t obligated to fast, and some say that they shouldn’t fast even if they want to.  Some poskim suggest that although there's no chinuch for the other 3 rabbinic fast days, for Tanit Ester there is an element of Chinuch to teach a child to fast on Tanit Ester.
- Someone who has some small illness but is in a lot of pain may eat on Tanit Ester, but should make the fast another day.
- What's the reason for Tanit Ester? The Shiltot (Shiltah 67) writes that both those who live in walled cities and those who live in non-walled cities should fast on the 13th of Adar because that was when the Jews gathered together to fast and pray prior to going to war in the days of Mordechai and Ester. Rabbenu Tam (cited by Rosh Megillah 1:1) agrees and supports this by saying that they must have fasted before going to war just as the Midrash says that the Jews fasted before the war with Amalek in the desert.
- On the other hand, the Kol Bo (Siman 62) explains that the fast is in commemoration of the fast that Ester decreed on the Jews of Shushan (Ester 4:16). Accordingly, the name “Tanit Ester” is fitting. The Avudraham (Taniyot s.v. VeAtta) argues that the fast Ester decreed was for 3 days and in Nissan, whereas our fast is only one day and in Adar. Rather he favors Rabbenu Tam’s reason.
- The Rosh Megillah 1:8 describes Tanit Ester as a rabbinic institution, while the Shibbolei HaLeket (Siman 194) quotes Rashi as saying that it is only a minhag. Hagahot Maimoniot (Taniyot 1:2) agrees. Based on the Rambam, the Mishna Brurah 686:2 rules that it is only a minhag. The Ran (Taniyot 7a s.v. Aval), however, quotes the Raavad who says that it is MeDivrei Kabbalah based on the words of “Divrei HaTzomot” (Ester 9:31). The Bach 686:2 writes that the Rambam agrees with the Raavad but concludes that most poskim hold it at most a rabbinic institution.
- Tosfot (Tanit 18a s.v. Rav) asks how we have the minhag to fast the day before Purim if according to the Megillah Tanit one may not conduct a eulogy on Purim and the Gemara (Tanit 15b) says that one may not fast on the day before the holidays on which one may not conduct a eulogy. Tosfot answers that Megillah Tanit was nullified with respect to the laws on the day before and after Purim and Chanuka. The Baal HaMa’or (Megillah 4a) answers that since Purim is considered Divrei Kabbalah, which is similar to Divrei Torah, there was no prohibition to fast on the day beforehand. The Meiri (Tanit 18a) quotes some who answer that that since Tanit Ester is a fasting of happiness, commemorating the miracle of the Jewish people’s success in war, it isn’t a violation of the prohibition to fast on the day before Purim.
- Interestingly, The Jewish Encyclopedia writes that the first to mention the first to mention this fast is the Shiltot (Shilta 67) in the Geonic period.
- The Shiltot (Shilta 67) writes that if Purim falls out on Sunday, they would fast on Thursday and not Friday because on Friday they would need to prepare for Shabbat. Tur and S”A 686:2 codify this as halacha. The Meiri in Magen Avot (Inyan 23) explains that although the fast days of mourning are delayed rather than pushed earlier, Tanit Ester is pushed up because it is a fast day of happiness.
- Sefer Eshkol 2:3 pg 7 and Bal HaMoar (Megilah end of first perek) write that it’s forbidden to fast on the 13th of Adar because it’s the day before a holiday. However based on many Rishonim, S”A rules 686:1 that one can fast the day before and after purim and so Tanit Ester should be observed as is Minhag Yisrael.
- Shiltot (Vayakhel 67) writes that the reason for Tanit Ester is to remember that the Jews fasted on the 13th of Adar (see Tanchuma Beresheet 3) in order to pray for mercy before the war they had with their enemies. Ravyah 550, Or Zaruha 2:437 pg 77d, Rosh (Megilah 1), Mordechai, Ran, and Tur 686 also quote this reason. However, Machsor Vitri 245 says that the fast is a minhag for the remembrance of the fast the Jews made in Nisan (mentioned in the Megilah) in days of Haman. This reason is also found in the Siddur Rashi 345, Shilbolei Leket 194 and Sefer Pardes 252. Sefer Manhig (Megilah 21) argues on this because we don’t fast 3 days or in Nisan. Actually, the Masechet Soferim (17:4, 21:1,12) says that since we are remembering the fast of Ester some have the Minhag to fast 3 fasts in Adar. This minhag is quoted in Or Zaruha, Rosh, Tur 686, and S”A 686:3. Sefer Eshkol (2:3 pg 7) and Magid Mesharim (Vayihakel) writes that the reason for the fast is so people listen in Megilah and not get caught up in eating. Ravad (quoted by Ran (Tanit 2nd perek)) and Sh”t Tashbetz 2:271 explain that the fast is based on the pasuk in the megilah “Kiymu Al Nafsham Divrei Tzomot”. Bach 686 says that the Ravad hold it’s a tanit that was established by the rabbis (in time of Ester) which is stronger than a Minhag. However Rambam (Tanit 5:5) writes that nowadays it’s only a Minhag (This is only true according to our (and the Bet Yosef’s) version of the Rambam but the Magid Mishna’s reading of Rambam leads one to say that perhaps the fast is an rabbinical establishment. Rama 686:3 concludes that the Tanit is only a Minhag and so one can be lenient for pregnant women.
- Halichot Shlomo (Moadim vol 2, 18:5), Rabbi Soloveitchik (cited by Rabbi Herschel Schachter in Nefesh Harav page 196)
- Nitei Gavriel (Purim 24:4, p. 165)
- The Mishna Brurah 550:6 quotes the Eliyah Rabba as saying that a righteous person (Baal Nefesh) should be strict to observe all of the 5 prohibited pleasures of Tisha BeAv on Shiva Asar BeTamuz, Asara BeTevet, and Tzom Gedalya. The Beiur Halacha 551 s.v. MeRosh Chodesh also quotes this. Rav Soloveitchik (cited in Nefesh Harav page 196) explained that this stringency doesn't apply to Tanit Ester, which is a fast of happiness and not mourning (see above). Rav Shlomo Zalman Auerbach (Halichot Shlomo 18:6) agrees.
- Levush 686:2.
- Shiltot (Vayihakel 67) and Tanchuma Beresheet 3 write that if Tanit Ester falls out on Shabbat it’s pushed earlier to Thursday. Rambam (Tanit 5:5), Tur and S”A 686:2 rule like the Shiltot. However, the Ravad (in Magen Avot 23 pg 152), quoted by Rashba (Eiruvin 41b), and Ritva (Eiruvin 41b)) holds that the fast occurs on Friday and one can eat after one prays right before Shabbat. Meiri (Magen Avot Inyan 23) proves that even a fast which is just a minhag is good enough reason to allow fasting onn Erev Shabbat and Erev yom tov from taanit bechorot on erev pesach. Sefer Pardes pg 252 says that some people would fast on Thursday with the Tzibbur and then fast again on Friday but says that they are mistaken.
- Sh”t Shevut Yacov 3:50 writes that if one mistakenly didn’t fast on Thursday can make it up by fasting on Friday. Shaarei Teshuva 686:5, Mishna Brurah 686:3, Torat HaMoadim 3:2 quote this as halacha.
- Mishna Brurah 686:3
- Concerning the 3 rabbinic fasts (Asara Betevet, Shiva Asher BeTamuz, Tzom Gedalya), S”A 554:5 (based on Magid Mishna (Tanit 5:10) and Rabbenu Yerucham 27) holds that pregnant and nursing women are exempt, while Rama 550:1 (based on Hagahot Maimon 5:1) holds they aren’t exempt only if they are in pain. Thus, concerning Tanit Ester, S”A will hold that these women are exempt. Additionally, the Rama 686:2 holds that these women are exempt even though they aren’t in pain, according to the explanation of Yeshuot Yacov 686:2 and Kaf HaChaim 686:18. Eshel Avraham MeButshatash 686, Sh”t Divrei Yacov Shor 41, Sh”t Maharsham 4:120, and Sh”t Hitorerut BeTeshuva 1:6(4) concur. Yet, some Achronim are strict within the opinion of the Rama including the Yavetz (Siddur Yavetz 26), Eliyah Rabba 686:2, Machsit HaShekel 686 and Chaye Adam 155:3. Mishna Brurah 686:4 quotes the dispute and in Shaar Tzion writes that one should follow the Minhag of the place.
- Concerning the 3 rabbinic fasts, Sh”t Ramah MePano 111, Sh”t Peni Yehoshua (2nd edition, O”C 16), Maharam Ben Chaviv in Tosfet Yom HaKippurim pg 30c write that a child is exempt. Chaye Adam 133:3 and Mishna Brurah 550:5 add that they are exempt from fasting partially (Tanit Shaot). Clearly concerning Tanit Ester, children are exempt and don’t have to fast partially. Torat HaMoadim 3:4 writes that even the Erech HaShulchan 554:2 who holds by the other fasts that a child should fast partially can agree by Tanit Ester that children are totally exempt.
- Rav Hershel Schachter in a shiur on yutorah.org ("Tanit Ester - Purim", min 15-20) explains that there's no chinuch to teach a child to fast for the 3 other rabbinic fasts because there's no chinuch for the laws of mourning and those fasts are fasts of mourning. However, Tanit Ester, is a fast of happiness, in remembrance of the fast of the Jews in the days of Ester, and so chinuch would apply to Tanit Ester.
- Darkei Moshe 686:1 quotes the Aguddah who says that someone who has an eye ache may eat on Tanit Ester but has to make it up another time. The Rama 686:2 rules that a Choleh Shein Bo Sakana or someone with an eye ache who is in great pain may eat on Tanit Ester but should make it up another day. Hilchot Chag BeChag (p. 29) generalizes this to say that anyone with a small illness who is great pain may eat but should make it up afterwards. The Aruch HaShulchan 686:4 writes that the Rama didn’t need to mention a Choleh Shein Bo Sakana except for those who held that Tanit Ester is MeDivrei Kabbalah, however, according to those who say it only a minhag, its obvious that a Choleh Shein Bo Sakana is exempt as he is even exempt from Tisha BeAv. Chazon Ovadya (p. 39) adds that anyone who has to eat on the doctor’s orders doesn’t have to make it up afterwards.