Learn Russian: У него есть какая-то тайна.

что-то means "something" and stands alone like a noun, while какая-то (from какой-то) means "some kind of" and acts like an adjective. Because it describes the noun тайна (secret), we need the adjective form. Since тайна is a feminine noun, we use the feminine form какая-то.

What exactly does the -то part do here?

The -то suffix turns a question word into an indefinite word. Without it, какая is a question word meaning "which" or "what kind of". By adding -то, it becomes an indefinite pronoun meaning "some" or "a certain". It tells us he has a secret, but we don't know what kind of secret it is.

Why isn't тайна in the accusative case, since it's the object he "has"?

In Russian, the "to have" construction (у ... есть) literally translates to "by [someone] there is [something]". The thing possessed (тайна) is actually the grammatical subject of the sentence, so it stays in the nominative (dictionary) case.

Why is there an "н" in у него? Isn't the word for "him" его?

Yes, the basic pronoun for "him" is его. However, Russian has a special rule: whenever a third-person pronoun (like him, her, or them) comes directly after a preposition (like у), you must add an н to the beginning of it to make it easier to say. That is why it becomes у него.