Learn Russian: Я знаком с этим человеком.

знаком is a short-form adjective meaning 'acquainted.' Even though it is not a reflexive verb like the ones in this lesson, it still describes a mutual relationship between two people, so it borrows the exact same pattern: с + Instrumental case.

If a woman is speaking, does the word знаком change?

Yes. знаком is a masculine short-form adjective. A female speaker would add an -а to the end and say Я знакома с этим человеком. If referring to a group ('we are acquainted'), it would be Мы знакомы.

Why is the word for 'this' spelled этим here?

Since the preposition с requires the Instrumental case, both the noun and the pointing word attached to it must change. The masculine dictionary form этот (this) changes to этим in the Instrumental case to match человеком.

Could I just say Я знаю этого человека instead?

Yes, you can! Я знаю этого человека uses the verb знать (to know) with the Accusative case and is very common. Saying Я знаком с specifically means 'I am acquainted with' or 'I have met,' which puts a little more focus on the two-way social connection.