Learn Farsi: مردی که ماشین شخصی داشت، به بیمارستان عمومی رفت. - A man who had a private car went to the public hospital.

These are relational adjectives, which are formed by taking a noun and adding the suffix ی (-i) to the end. شخص means person, so شخصی means personal or private. Similarly, عموم means the public, so adding ی creates عمومی (public or general).

The word مردی (a man) also ends in ی. Is that the same adjective suffix?

No, and this is a great catch! While they look and sound exactly the same, the ی in مردی is the indefinite article meaning a or an (مرد + ی = a man). The ی at the end of شخصی and عمومی is a completely different suffix used to turn nouns into adjectives.

How do we connect the adjectives to their nouns here, like private car?

You must use the ezafe, which is a short e sound that connects a noun to its adjective. It usually isn't written in the standard alphabet, but it is spoken. You pronounce private car as ماشینِ شخصی (māšin-e šaxsi) and public hospital as بیمارستانِ عمومی (bimārestān-e omumi).

How does the word که work in this sentence?

Here, که acts as a relative pronoun linking the man to his description. Since it comes right after مردی (a man), it translates to who (a man who...). is a very handy word in Farsi because it can mean who, which, or that depending on the noun it is describing.