BeHaB begins this coming Monday, May 13th. BeHaB is a series of three fast days on the Monday, Thursday, and subsequent Monday following the months of Tishrei and Nissan (BeHaB is an acronym for Beis (Monday), Hei (Thursday), Beis (Monday)). A Mi Shebeirach is recited before Mussaf this Shabbos for those who fast on these days. Many congregations recite Selichos on these days.
A longstanding custom dating back to the era of Tosafos (Kiddushin 81a) involves fasting after Pesach and Sukkos.
This practice is based on several reasons:
The Tur (492) mentions that since the Chagim are periods of joy and feasting, there's a concern that it may lead to transgressions, so the fast is observed as an atonement. Tosafos (ad loc.) highlights that potential sins could involve inappropriate interactions between men and women during the festivities.
Levush (492) suggests fasting can benefit one's health as the seasons transition, safeguarding against potential risks associated with seasonal changes.
The Eliya Rabba (492,3) says that since it's easy for a person to make mistakes with the Halachos of Yom Tov, like working on Chol Hamoed, cooking on Yom Tov when not permitted, etc., we fast to atone for those sins.
These days, it's difficult for many to fast, and it will affect one's living and Torah study, so many don't fast. It's important to note that Piskei Teshuvos (492,1) writes that even if one chooses not to fast, indulgence in delicacies during these days should be avoided.