الْحَمْدُ
لِلَّهِ رَبِّ الْعَالَمِينَ
رَبِّ
الْعَالَمِينَ
رَبِّ
The word is in the genitive (مجرور) state because it is the adjective (النعت), and according to some the appositional substantive (البدل), for Allah.
The adjective takes on the same vowel signs as its noun. Since, Allah has kasrah under the last letter, so does the adjective, Rabb(i).
The word is also forming a construct (المضاف والمضاف إليه) with the word which follows. It is the (مضاف) and the word which follows is its (المضاف إليه)
The word which follows is
الْعَالَمِينَ
This word is in the genitive (مجرور) state because it is the second member of the construct.
The second member of a construct (المضاف والمضاف إليه), known as the (المضاف إليه), in in the genitive state (مجرور)
The word is a sound masculine plural (جمع المذكر السالم) and hence its genitive state (مجرور) is indicated by the Yaa (الياء) before the Noon (النُّونُ)
Rabb means Lord
Al-'aalameen means the worlds, all that exists
Hence, we have
Lord (of) the worlds or Lord (of) all that exists