This is the 859th most frequent Russian word.
европейского
"European" (adjective, relating to Europe).
In this sentence, "европейского" is a genitive singular masculine/neuter adjective form, describing "искусства" (art) as being related to Europe, indicating the origin or style.
Мы посетили музей европейского искусства.
We visited the museum of European art.
Here, "европейского" describes "строительства" (construction) in the genitive singular masculine/neuter form, identifying the construction style or characteristics as European.
Это один из самых красивых проектов европейского строительства.
This is one of the most beautiful projects of European construction.
"Европейского" modifies "региона" (region) and is in genitive singular masculine/neuter form, denoting that this specific region is associated with Europe.