Sick or weak fasting on Rabbinical fast days :

Must one fast if he is sick:^[1] One who is sick is exempt from fasting, [and is even prohibited to be stringent upon himself to fast[2]]. This applies even if there is no danger involved in his illness.^[3] He is not required to make up the fast on a later date when he feels healthy.[4]

One who is not sick but is weak: One who is weak is not required to fast if he receives a medical order from a doctor that the fast is detrimental to his health.^[5] He is not required to make up the fast on a later date when he feels healthy.[6] If the fast causes one to become extremely weak, more than the average person, some Poskim[7] rule he is considered like a sick person and is not required to fast. 

One who is not sick but is in pain: One who is not sick must fast even if he is experiencing a great amount of pain.[8] However, some Poskim[9] rule one may be lenient if he is in a great amount of pain.

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]

Taanis Nidche:[11] If the fast fell on Shabbos and was hence differed to Sunday, one may be lenient even if he is a slightly sick, and is not bedridden.

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^[1] Rama 686:2 [regarding Taanis Esther]; Chayeh Adam 133:6; M”B 550:4; Kaf Hachaim 550:7

[2] M”B ibid; Kaf Hachaim ibid

[3] Chayeh Adam 133:6; M”B 550:4; See Q&A for definition of sick

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

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

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

[7] Ashel Avraham Tinyana 550; Igros Moshe 4:114

[8] See Rama 686:2 and Kaf Hachaim 686:17 who 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.

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

[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 Eiger 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] So rules regarding even Tisha B’av Nidche: Shvus Yaakov 3:37; Hagahos Rav Akiva Eiger 559:9; Nitei Gavriel 5:9