This is the 6982nd most frequent Russian word.
архангельская
"Архангельская" means "Arkhangelskaya," referring to something related to Arkhangelsk, a city or region in Russia.
Here, 'Архангельской' is an adjective modifying 'области' meaning 'region', referring to the specific territorial region of Arkhangelsk.
В Архангельской области очень красивая природа.
In the Arkhangelsk region, the nature is very beautiful.
'Архангельский' here functions as an adjective derived from 'Архангельск', used to describe something characteristic of or related to Arkhangelsk.
Её стиль был почти архангельский.
Her style was almost Arkhangelsk-like.
In this case, 'Архангельскую' is an adjective modifying 'кухню' (cuisine), referring to the specific culinary tradition of Arkhangelsk.