Learn Hardcore Turkish: Eski pasajın tavanında muhteşem vitray pencereler var. - There are magnificent stained-glass windows on the ceiling of the old passage.

-ın is the genitive suffix in Turkish. When you want to express “of something” (possession), the owner noun takes a genitive ending.
• pasaj → pasaj + ın = pasajın (“of the passage”)

Why does tavanında have two suffixes and what do they mean?

The word tavan (“ceiling”) carries:
1) -ı (3rd-person singular possessive) → tavanı = “its ceiling”
2) -nda (locative) → tavanında = “on its ceiling”

So Eski pasajın tavanında literally means “on the old passage’s ceiling.”

What function does var serve in this sentence?

var is the existential verb meaning “there is / there are.” In Turkish existential sentences the pattern is:
(locative) + (subject) + var/yok.
Here, muhteşem vitray pencereler is the subject, and var tells you that they exist.

Why is vitray singular while pencereler is plural?