I think the first is more common. An example of the second used in context is, "I haven't understood anything at all that the professor has been lecturing about, so I don't think I'm going to pass this course." -------- I don't know how helpful my answer is to you. I guess it's an example of Daniela's explanation.

It depends on the context, meaning the period of time where the action took place. I did not understand: past tense, meaning the action is over. I have not understood: present perfect, meaning you did not understand and you do not understand yet.

i guess its better: I don't/didn't understand :)

If you were sitting in class, and the professor says something that you didn't understand, one might probably say, "I'm sorry, I didn't quite understand that. Can you explain it to me again.?"