The מחבר (סימן תקמ”ט סעיף א’) brings the halacha of the four days that one must fast. One of them is the tenth of Teves. On this day Nevuchadnetzer the King of Bavel laid siege on Yerushalayim which eventually brought about the destruction of the Bais Hamikdash.
There is a machlokes Rishonim as to the purpose of these fast days. The Rambam says they were established by the Neviim to be days of repentance. The Ritvah and Ran however are of the opinion, that they are days of mourning for the terrible tragedies that transpired on those days. The Mishnah Berurah brings the opinion of the Rambam that they are days of teshuva. Therefore one is obligated on these fast days to search his ways and do teshuvah. The fasting is not the main idea; rather it is only a preparation to actually do teshuva. Someone fasting who goes on leisure strolls, wastes his time, and does not do teshuva, is not fulfilling the main reason of the fast day.
The Mishnah Berurah (סימן תקס”ו ס”ק י”ב) says there is a minhag to give out tzedakah at mincha time on the day of the fast. The source for this is the gemara in Berachos (6b) which says אגרא דתעניתא צדקתא – the reward for a fast day is in exchange for giving tzedakah. Rashi explains that the poor people will have something to eat after the fast is over. The אליה רבה says that there are those who give to tzedakah, the amount that they would have eaten that day.
The מחבר (סימן תק”נ סעף א’) writes that everyone is obligated to fast on these four days “ואסור לפרץ גדר” it is prohibited to breech this takanah. However the Rema adds that if pregnant and nursing mothers will be in a lot of pain if they fast, then they should refrain from fasting. Then he brings a more lenient opinion that even if they are not in pain they are not obligated to fast, still nevertheless the women are stringent on themselves and do fast.
The Mishnah Berurah writes that it is most probable that she is considered pregnant when it is visible to the eye just as in hilchos niddah. Nevertheless if it is within the first forty days and she is feeling צער, she has the status of a pregnant woman. Before forty days however, she is like all other women unless she has a lot of צער. If a pregnant woman is feeling very weak she should not fast. This is especially true if it is during the beginning of pregnancy, since it can cause a miscarriage and could also harm the mother.
In Hilchos niddah a woman is considered a מינקת for twenty four months even if she is not nursing. Most poskim say that this does not apply to the halachos of fasting. Therefore, even if she is still within twenty four months of having a child she must fast. If she is only partially nursing, the Chazon Ish holds that she is still considered a מינקת. If she is feeling weak some poskim allow her not to fast. The דעת תורה is of the opinion that even if she is not nursing at all she is still considered a מינקת and does not have to fast.
The Chazon Ish zt”l holds that a pregnant and nursing woman should start the fast, and when they start to feel pain or weakness only then they should start to eat. Reb Nissim Karelitz zt”l rules that for the fast days during the summer time she does not have to even start the fast. For עשרה בטבת which is during the winter time, she should start the fast. The אור לציון however rules that today all pregnant and nursing women are מצטער and do not have to fast at all. All Poskim agree that if she will not be able to nurse her child if she fasts then she should not fast at all. When one is permitted to eat, she should only eat what she needs. She should not indulge in meat, wine and delicacies.
Any woman who is not מעוברת או מינקת and is not a חולה is obligated to fast. In the Chassidic circles many are lenient in regard to women fasting. This is also the practice of Brisk וצ”ע.
Prepared by R’ Avrohom Yehoshua Ziskind
 Sources:משנה ברורה,דרשו,פסקי הלכות