when should I add "it" after believe or understand?

Solution 1:

Like any other context, we use the pronoun "it" to refer to a specific thing previously mentioned or easily identified.

Let's say that you ask somebody to explain algebra to you, because you don't understand algebra. If, after they had explained you said "I don't understand it", as the subject you are speaking about is algebra, it would be taken that "it" refers to this and that you still don't understand algebra.

However, if you simply said "I don't understand" this could mean that you didn't understand the way it was explained to you, or that you don't understand something new which has been brought up during the explanation. Maybe the person replied in a language you didn't understand!

It isn't particularly idiomatic to say "I don't believe" without directing this at something - "I don't believe it" or "I don't believe you".