Learn Turkish: İspanyolca sertifikası aldığın okul nerede? - Where is the school where you got the Spanish certificate?

In English, we use "where" to link a place to a relative clause. Turkish doesn't use a separate connecting word for this. Instead, it uses the exact same -dık / -dik participle that it uses for direct objects. Because a school is a place, context tells us that aldığın okul means "the school where you got [something]" rather than "the school that you bought."

How do we know it was you who got the certificate, since the word senin (your) isn't in the sentence?

In this type of relative clause, the person doing the action is marked by a possessive suffix on the participle itself. The base participle is aldık (getting). By adding the 2nd person singular possessive suffix -ın, it becomes aldığın (literally "your-getting").

Why does sertifika have an -sı at the end?

This is a noun compound (two nouns joined together to make a new concept). When the first noun doesn't actually "own" the second noun, only the second noun gets a possessive suffix. İspanyolca (Spanish) + sertifika (certificate) + -sı (compound suffix) makes İspanyolca sertifikası (Spanish certificate).

Since the certificate is the object being received, why isn't it in the accusative case (e.g., sertifikasını)?