Rajab (ِرجب) is the seventh month of the Islamic calendar. The lexical definition of the classical Arabic verb rajaba is "to respect" which could also mean "be awe or be in fear", of which Rajab is a derivative. This month is regarded as one of the four sacred months in Islam in which battles are prohibited.
Thank you for your question. It is recommended to fast for both the months of Rajab and Shaban.
May you always be successful
Yes of course. In fact it is mentioned in many Hadeeths from the Prophet Muhammad (SAWA) and his Holy Progeny Ahlul Bayt (AS) that it is recommended to fast any number of days or the whole month of Rajab and any number of days or the whole month of Sha'ban for those who can, and for those who can not fast the whole month, then to fast any number of days in Rajab and Sha'ban. The Prophet (SAWA) himself used to fast the whole month of Sha'ban and used to link it with fasting of the month of Ramadhan.