The Mishna in Masechet Ta’anit says that in ancient times, when the Rabbis would occasionally declare a series of fasts in times of crisis, the fasts were observed on Monday, Thursday and Monday. The series of fasts always began on a Monday, with the second fast observed the Thursday after that, and the third on the following Monday.
The Gemara explains that the series of fasts does not begin on Thursday because this could affect prices in stores. If the first fast would be observed on Thursday, people would go to the stores on Thursday night to purchase food for breaking their fast, and also for Shabbat. The shopkeepers, who may not have heard of the declaration of a series of fasts, might wrongly assume that the people are stocking up because of a food shortage, and they will then raise the prices. The Sages therefore enacted that the series of fasts should begin on Monday, so that people would shop normally on Monday night, and by Thursday night the shopkeepers would certainly have received word of the declaration of fasts.
The Gemara does not, however, explain why specifically Monday and Thursday were chosen as fast days. It works off the assumption that these days are appropriate days for fasting, and then discusses the reason why the first fast is observed on Monday instead of Thursday. But why are these days of the week especially suited for fasting?
This question was addressed by Tosafot (commentaries by French and German Medieval scholars), in Masechet Bava Kama (82), and they explain that Monday and Thursday are "Yemeh Rason" – days when our prayers are more readily accepted by G-d. After the sin of the golden calf, Moshe ascended Mount Sinai to receive the second tablets. According to tradition, he ascended the mountain on a Thursday and descended on a Monday. This was the time when G-d mercifully forgave the people for the grave sin of worshipping a calf, and the fact that this process began on Thursday and ended on Monday establishes these days as special days of divine mercy and compassion. Hence, when choosing days for fasting and prayer in times of crisis, the Sages instituted that the fasts be observed on Monday and Thursday.
The Rosh (Rabbenu Asher Ben Yehiel, 1250-1327), in Masechet Taanit, adds that this is also the reason why we add special Tahanunim to our morning prayer service on Mondays and Thursdays. As these days are earmarked for compassion and acceptance of prayer, we want to take advantage of this opportunity and we add special supplications. And thus even though we do not usually observe extra fasts nowadays, we should be aware that Mondays and Thursdays are especially suited for having our prayers answered, and we should therefore endeavor to pray with particular feeling and concentration on these days.