In Arabic, every one has a name like Husain and a Kunyah like Abu or Aba or Abi. The name of the Prophet (SAWA) is Muhammad and his Kunyah is Abu Qasim. Ameerul Mo'minen (AS) name is Ali and his Kunyah is Abul Hasan. Imam Hasan's Kunyah is Abu Mohammad. mam Husain's Kunyah is Abu Abdillah or Aba Abdillah.
Abu Abdullah means father of Abdullah. Imam Husain had a son named Abdullah.