Could you tell me which is proper in English:
I don't remember.
I can't remember.
And if both are possible and good English, what is the difference?
They are both good English, and the difference in actual practice is very slight.
I don't remember means that I once knew, but now I do not. This is less committal, since it sounds as if I am not necessarily trying very hard to remember.
I can't remember means that I once knew, but now I cannot think of it. This is more emphatic, since it sounds as if I am testing the limits of my memory.
They are both good English.
The only difference between them (in my dialect) is that I can't remember indicates that the speaker tried to remember, whereas I don't remember doesn't imply that.
Thus: Can you remember what happened that day? is asking the addressee to try to remember, whereas Do you remember what happened that day? is merely asking him whether he happens to remember.
(This answer is similar to Daniel δ's, but I'm not sure he means what I do.)