This is the 12931st most frequent Russian word.
ливанской
"Ливанской" means "Lebanese" in English, relating to Lebanon.
In this sentence, 'ливанской' is used as an adjective in the genitive form to describe 'кухни' (cuisine) as being Lebanese.
Я обожаю блюда ливанской кухни.
I adore dishes of Lebanese cuisine.
Here, 'ливанской' serves as an adjective in the instrumental form to specify 'культурой' (culture) as Lebanese.
Она увлекается ливанской культурой и традициями.
She is interested in Lebanese culture and traditions.
'ливанской' in this context is an adjective in the genitive form describing 'керамики' (ceramics) as Lebanese.