Learn French: Je dois étudier pour réussir mon examen.

In English, we often just say "to pass" to mean "in order to pass". In French, when you want to show the purpose or goal of an action, you use pour followed by the infinitive (unchanged) verb. So, pour réussir translates closely to "in order to succeed".

I thought "to pass" translated to passer. Why use réussir?

This is a classic false friend! In French, passer un examen actually means to take an exam. If you want to say that you got a good grade and passed it, you must use réussir (which literally means "to succeed").

Why isn't étudier conjugated to match Je?

Because it comes directly after another verb that is already conjugated. Je dois means "I must" or "I have to". When two verbs go together like this, the second one stays in its dictionary form (the infinitive).

Why do we use mon for "my" here?

The word examen is a masculine noun in French. To say "my" with a masculine singular noun, you use mon.