The month after Ramadan is Shawwal. And the month before Ramadan is Shaban. Shawwal and Shaban are the months before and after Ramadan. Therefore, in honor of Ramadan, one should be careful to fast in these tow months, especially in the month of Shawwal, because it is encouraged in Hadith. Abu Ayyub Ansari r. said, the Prophet, may Allah bless him and grant him peace, said:

مَنْ صَامَ رَمَضَانَ ثُمّ أَتْبَعَهُ سِتّا مِنْ شَوّالٍ، كَانَ كَصِيَامِ الدّهْرِ

One who fasts in Ramadan then fasts six days in Shawwal, it will be equivalent to fasting the whole year. -Sahih Muslim, Hadith 1164; Jame' Tirmidhi, Hadith 759; Sunan Abu Dawud, Hadith 2433

In another hadith, the Prophet sallallahu alaihi wa sallam said,

صِيَامُ شَهْرِ رَمَضَانَ بِعَشَرَةِ أَشْهُرٍ وَصِيَامُ سِتّةِ أَيّامٍ مِنْ شَوّالٍ بِشَهْرَيْنِ فَذَلِكَ صِيَامُ سَنَةٍ

Fasting in Ramadan is equivalent to ten months and fasting in six days of Shawwal is equivalent to two months. So, Ramadan and six days of Shawwal together are equal to one year. -Sunan al-Kubra, Nasa'i, Hadith 2873; Sahih Ibn Khuzaimah, Hadith 2115; Musnad Ahmad, Hadith 22412; As-sunan al-Kubra, Baihaqi 4/293

In another narration of the hadith-

جَعَلَ اللهُ الْحَسَنَةَ بِعَشْرٍ فَشَهْرٌ بِعَشَرَةِ أَشْهُرٍ وَسِتّةُ أَيَّامٍ بَعْدَ الْفِطْرِ تَمَامُ السّنَةِ

Allah has made one good deed equal to ten good deeds. So one month (of Ramadan) is equal to ten months and fasting for six days after Eid-ul-Fitr is equivalent to tow months. Thus, it is equal to fasting for the whole year. -Sunan Nasa'i, Hadith 2874

Allah gives ten rewards for one good deed. Fasting one day is equal to fasting ten days, fasting three days is equal to fasting one month. Fasting six days is equal to fasting two months. Fasting one month is equal to fasting ten months. So whoever fasts one month of Ramadan and six days of Shawwal will get a reward equal to one year.