I have got really confused by the following two sentences.
I don't remember the name
I can't remember the name
As I understand, the second one means:
You are not able to recall the name from your memory.
But what does the first one means?
They mean the same, but only because the word "remember" means "be able to recall from memory"
As the word remember includes the sense "be able to..." so "I don't remember" and "I can't remember" mean the same. You might use "I can't remember" to mean "I've tried to remember and failed" but "I don't remember" to mean "I don't know, but I haven't thought hard about it yet". But really this is only nuance and not strictly followed.
Hello again, umm — I'm really sorry, I can't remember your name. (I'm using "can't" to suggest that I've really tried, but "don't" would also be correct.)
I met a girl at the bar yesterday, I don't remember her name, but she was wearing a long red dress. ("can't would be perfectly accceptable here.)