I don't understand vs I didn't understand
When having a conversation, what kind of context is more appropriate to use "I don't understand." / "I didn't understand"?

Question

Answer

This depends on what you are referring to - is it something current or in the past?

I don't understand is present tense. 
You could say, "I don't understand what you are [currently] saying"

I didn't understand is past tense.
You could say, "I didn't understand what you said five minutes ago"

Answer

I don't understand = now
I didn't understand = past

I don't understand your question.

I didn't understand your question (before) but someone helped me (to understand), so I do understand (it) now.