Not quite. I'm pretty certain that "Al" in Arabic means "from" in English, so "Al-Ghamdi" would mean "from Ghamdi", as in a city or district called Ghamdi, sorta like Occam of Occam's Razor fame was William of Occam, as in, from the British city of Occam.
I don't speak Arabic, so take this with a grain of salt, but so far as I know "Ali" means "Great" and Adbul means "slave", so Ali Abdul Rachman would be "Great Slave Rachman", and Ali Abdul Rachman Al-Gamdi would be Great Slave Rachman from Gamdi.
Likewise, since "Abu" means "father" then "Abu Bakr al-Azdi" would mean "Father of Bakr from Azdi".
So this is how so many Muslims have more than one name. The guy has his childhood name and then he has his post-marital name. He's not just Great Slave Rachman born in Gamdi, but now he is the Father of Bakr who was born in Azdi.