When is the fasting of Ayyam al-Bid (the white days) observed and what is its significance?

Ayyam al-Bid (the white days) are the 13th, 14th, and 15th nights of the lunar months, when the moon is at its brightest. (See: Bukhari, Sawm, 60 [1981]; Nasa'i, Siyam, 84 [2432]) These nights are called so because the moon appears in full and illuminates the nights more than usual. The Messenger of Allah (peace be upon him) recommended fasting on these days each month (Abu Dawud, Sawm, 68 [2449]; Tirmidhi, Sawm, 54 [761]) and stated that fasting on these days is like fasting the entire year. (Abu Dawud, Sawm, 68 [2449]; Ibn Majah, Siyam, 29 [1707])

Source: T.C. Diyanet İşleri Başkanlığı, Din İşleri Yüksek Kurulu