Those exempt from the fast:^[1]
A. Sick or weak:
One who is sick is exempt from fasting. This applies even if there is no danger involved in his illness.^[2] It is forbidden for him to be stringent.[3] One who is old or weak is not required to fast if he receives a medical order from a doctor that the fast is detrimental to his health.^[4] He is not required to make up the fast on a later date when he feels healthy.^[5] One who is in great pain, or will suffer due to the fast more than a regular person, is nevertheless required to fast.[6] However, some Poskim[7] rule one may be lenient in such a case. If the fast causes one to become extremely weak, more than the average person, some Poskim[8] rule he is considered like a sick person and is not required to fast. 
One who is traveling:^[9] If one feels healthy he must fast even if he is traveling that day.

Q&A How sick must a person be to be exempt from the fast? One must be sick enough that he feels weak in his entire body due to the illness, or is bedridden.[10]  Must an old or weak person fast?[11] An old person, or a person who is weak, is not to fast on Taanis Esther or on any of the three Rabbinical fasts.[12] Practically, however, one may not be lenient unless he receives a medical order from a doctor that the fast is detrimental to him.  May one take medicine on a fast day?[13] If one is sick then he is not required to fast, as explained above. Even if one is not sick, it is permitted to take medicine if the medicine is bitter. He is to swallow it without water or alternatively place something bitter in the water and use this water to help swallow the pill. [If one is unable to do so, then the medicine may be taken if the person will fall sick if he does not take it.[14]]

 B. Pregnant or after birth:^[15]
Pregnant^[16] and nursing^[17] women are exempt from fasting. Nevertheless the custom is for even pregnant and nursing women to fast unless they are in great pain [or feel weak^[18]] in which case they are specifically not to fast. They are likewise not required to make up the fast at a later date. [Practically the widespread custom today is for pregnant women not to fast as they are all considered weak.^[19] They are also not required to make up the fast at a later date.^[20]]
Women after birth:^[21] According to all opinions a woman within 30 days of giving birth does not need to fast.

Q&A From what stage in pregnancy is a pregnant woman exempt from fasting?[22] A pregnant woman that is in pain[23] is exempt from fasting starting from the 41st day of her pregnancy. If she feels weak or in great pain then she may be lenient even prior to 40 days. If she does not feel weak or any pain at all then she is required to fast if she has not yet entered her second trimester. Must a woman which is after a miscarriage fast?[24] No. Must a woman within two years of birth fast? Some Poskim[25] rule that all women within 24 months after birth [child is less than 2 years old] are defined as nursing women, even if they are not actually nursing anymore

B. Children:
All children that are above the age of Bar or Bas Mitzvah are obligated to fast.^[26] All children below the age of Bar or Bas Mitzvah are not required to fast any of the four Rabbinical fasts.^[27] One is not even required to educate the child to fast for a certain amount of hours into the day.^[28] Some Poskim^[29] however rule that a child which has reached the age of understanding the mourning^[30] is only to be allowed to eat bread and water or other simple staple food. The custom however is not like this opinion.^[31] [However they are not to be given sweets and the like although one is not required to stop them from eating it if they are in the midst of doing so.^[32]] 

Q&A Must one who needs to eat or drink only take the minimal measurements “Shiurim” as required on Yom Kippur? Some Poskim[33] rule one is required to follow the same eating and drinking measurements “Shiurim” prescribed on Yom Kippur. Other Poskim[34] however rule it is not necessary. Practically, the custom is not to follow this opinion, and hence one who is permitted to eat or drink may do so without limitation.[35] Must a Chasan and Kallah within the 7 days of Sheva Brachos fast?[36] Yes. May one who is not fasting eat like a normal day?[37] It is proper to diminish somewhat in eating and drinking if one is not fasting that day, such as a pregnant or nursing woman, in order to participate in the day of oppression followed by the community. Must a soldier in the army fast? If he is not involved that day in any military operation or warfare then he must fast like everyone else. If however he is taking part in a military operation then he must eat in order to have strength. May a kosher restaurant open on a fast day?[38] Yes. However they may only sell food to Jews that do not have to fast. May one touch foods on a fast day?[39] It is permitted to touch food on a fast day.[40]

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^[1] 550/1

[2] Chayeh Adam 133/6; M”B 550/4; Kaf Hachaim 550/7;

[3] M”B ibid; Kaf Hachaim ibid

[4] Ruach Chaim 550/1; Kaf Hachaim 550/6; However without a direct medical directive he is not to be lenient. [ibid]

[5] See Rama 586/2; M”B 686/5; Shaareiy Tziyon 586/11; Kaf Hachaim 686/22

[6] See Rama 686/2 and Kaf Hachaim 686/17 that implies the leniency of pain is only by Taanis Esther, and so suggests also Biur Halacha 550/1 “Mihu”; Kaf Hachaim ibid writes that according to those who rule the fast is an obligation certainly one who is in mere pain may not be lenient to eat, unless he is sick, just as is the law by the other four mandatory fasts of scripture.

[7] Biur Halacha ibid in name of Kovetz on Rambam; Sdei Chemed Mareches Bein Hametzarim 1/13; Ashel Avraham Tinyana 550; Daas Torah 550; Nitei Gavriel 5/7; See also Igros Moshe 4/114

[8] Ashel Avraham ibid; Igros Moshe 4/114

[9] M”A 668/4 brought in M”B 686/6

[10] As this is considered a Choleh Sheiyn Bo Sakana, mentioned in M”B 550/4; and so is implied also from Aruch Hashulchan 550/7; and Hagahos Rav Akiva Eoger 559/9; See however Aruch Hashulchan 554/7 that this applies even if he is not bedridden; See Nitei Gavriel 5/4 in name of Maharam Bent 550 that even if he is slightly ill one may be lenient

[11] Kaf Hachaim 550/6 in name of Ruach Chaim 550/1

[12] The reason: As they are no different than a pregnant or nursing woman who is exempt from fasting. [Kaf Hachaim ibid]

[13] Kaf Hachaim 554/34 in name of Kesonos Yosef 4; Ikarei Hadaat 29/36; Tosefes Chaim on Chayeh Adam 1/135-8; ; Piskeiy Teshuvah 567; Pischei Olam 554/6; Piskeiy Teshuvos 554/7; 568/3

[14] Igros Moshe 3/91; Piskeiy Teshuvos ibid

[15] Rama 550/1

[16] See Q&A

[17] Some Poskim rule that all women within 24 months after birth [child is less than 2 years old] are defined as nursing women, even if they are not actually nursing anymore. [Daas Torah 550; Ashel Avraham Tinyana 550/1; Alef Hamagen 3; Piskeiy Teshuvos 550/1; See Nitei Gavriel Bein Hametzarim 5/1 footnote 1] See however Nitei Gavriel ibid for Rabbanim who disagree with this ruling.

[18] M”B 550/5

[19] See Piskeiy Teshuvos 550/1; Piskeiy Teshuvos 686/3 that so is the ruling of today’s Rabbanim. So rules Ashel Avraham Butchach 686 Mahadurah Tinyana; Hisorerus Teshuvah 1/6; Divreiy Yatziv 2/291

[20] M”B 686/5; Kaf Hachaim 686/22 regarding Taanis Esther and the same would apply regarding the four fasts; See however M”B ibid in name of the Bach.

[21] See M”B 686/4; Kaf Hachaim 686/19

[22] M”B 550/3; Kaf Hachaim 550/5; Mor Uketzia 550; Piskeiy Teshuvos 550/1; Nitei Gavriel Bein Hametzarim 5

Background:

The M”B ibid rules that seemingly she is considered pregnant starting from the 4^th month and onwards, however possibly she is considered pregnant even starting from the 41^st day if she feels pain. However if she is pregnant for less than 40 days she is considered like any other women that must fast unless she is in great pain. He then concludes that if she feels weakness she is not to be stringent at all. The Kaf Hachaim ibid however writes this ruling a little differently: He writes that if she is in great pain she may be lenient from the 41^st day, however prior to 40 days she must fast even if she is in great pain. 

[23] See Background!

[24] Piskeiy Teshuvos 550/1; There he brings that they are exempt from fasting for a full 24 months after the miscarriage just like a woman who gave birth.

[25] Daas Torah 550; Ashel Avraham Tinyana 550/1; Alef Hamagen 3; Piskeiy Teshuvos 550/1; See Nitei Gavriel Bein Hametzarim 5/1 footnote 1] See however Nitei Gavriel ibid for Rabbanim who disagree with this ruling.

[26] Admur 616/9; Elya Raba 549/7; Kaf Hachaim 550/1

Two hairs: The above law only applies if the child has reached puberty which means that he or she has grown two pubic hairs. If the child has reached this age but does not have two pubic hairs, then although he or she must keep all the commands out of doubt that perhaps he or she grew the hairs and they fell off, nevertheless the child is not required to fast any of the Rabbinical fasts. [Admur ibid]

Other Poskim: Some Poskim rule the child must fast even the Rabbinical fasts once he or she reaches 13/12 even if the child has not yet grown two hairs. [P”M 550 A”A 2; Biur Halacha 550 “Hakol”; Kaf Hachaim ibid]

[27] Admur 616/9; Elya Raba 549/7; M”B 550/5; Kaf Hachaim 550/1

[28] Chayeh Adam 133/6; M”B ibid; Kaf Hachaim 550/9; Piskeiy Teshuvos 550 footnote 10

Other Opinions: Some Poskim rule that children [of nine years old] are to be educated to fast for a few hours into the day just as is the law on Yom Kippur as rules Admur 616/5. [Erech Hashulchan 554/2 and Beis Hillel brought in Kaf Hachaim 554/23; Chanoch Lenaar 21 footnote 9; Beir Moshe 8/98; Piskeiy Teshuvos 554/9]

[29] M”A 550/2; Rameh 111; P”M 549 A”A 7; Chayeh Adam ibid; M”B ibid; brought in Kaf Hachaim 549/8 and 550/9

[30] Some write this is starting from age nine. [See Piskeiy Teshuvos 550 footnote 11]

[31] Birkeiy Yosef 549/1; Daas Torah; Kaf Hachaim 549/8 and 550/9; Beir Moshe 8/95

[32] Beir Moshe ibid; Piskeiy Teshuvos 550/3

[33] Tzemach Tzedek O.C. 108/110; Divrei Nechmia 42; Sefer Haminhagim p. 92 [English] regarding all four fasts and in footnote 342 even regarding Taanis Esther; Halachos Ketanos 2/100; Mahrahm Shick 289; Shaiy Lamorah 4; Marcheshes 1/14; Biur Halacha 554 “Bemakom” in name of Pischeiy Olam

[34] Chida in Machazik Bracha, brought in Kaf Hachaim 554/31; Mahrahm Shick 290; Avnei Nezer 540; Aruch Hashulchan 554/7

[35] Shevet Halevi 4/56; Kinyan Torah 1/118; Tzitzi Eliezer 10/25-16; Shraga Hameir 1/59; Piskeiy Teshuvos 554/9; Hiskashrus 889 that so he received from several Rabbanei Anash, despite the ruling in Sefer Haminhagim ibid

Some suggest the entire reason behind the Shiurim recorded in the previous Poskim who are stringent is order to also gain the advantage of being considered to have fasted, and hence be able to count for a Minyan for Kerias Hatorah, and saying Aneinu and the like, however not that one is required to follow these Shiurim. [See Hiskashrus ibid; Piskeiy Teshuvos ibid and 566/6; Marcheshes ibid]

[36] Ritva Taanis; Peri Hadamah 4/41; M”A 550/5; Kesones Yosef 5; Beis Yehuda 31; Birkeiy Yosef 549/2; Kaf Hachaim 549/9

The reason: As the mourning of the public differs the Holiday of an individual. [Ritva ibid]

Other Poskim: Some Poskim are lenient in this matter and rule a Chasan is not required to fast, as his Yom Tov is greater than the Yom Tov of Mila and hence pushes off Aveilus. [Beis David 476 brought in Kaf Hachaim ibid]

[37] Ashel Avraham 686 Mahadurah Tinyana

[38] Piskeiy Teshuvos 549/1

[39] Admur 611/9; Rama 612/10

Other opinions: Some Poskim rule it is forbidden to touch food unnecessarily during a fast day, lest one come to eat the food. [P”M 612 M”Z 10 based on Taz 612/8 and M”A 612/6; Kaf Hachaim 612/56]

[40] The reason: As since one refrains from eating any food that entire day, he will not come to accidently eat the food that he touches, unlike the law of Chametz on Pesach. [Admur ibid]

Making food: The above law in 611/9 is regarding touching food, however in 611/7 and 9 Admur rules regarding Yom Kippur that one may not make food lest he come to eat it. Vetzaruch Iyun if this stringency applies during the four fasts, as learns the P”M ibid. Practically the custom is to be lenient.