What is the difference between 나, 난, and 나는??

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나 is 1st singular.
It can both mean I and me
so you have to put subject marker or object marker in order to make it clear whether 나 is used as a subject or object.

는 is one of the subject markers.
so, 나+는 becomes 나는 and it means I
eg. 나는 학생이다
I am a student

나는 is often contracted to 난, which is similar case with English, like we say I'm instead of I am.

When 나 is used without any subject or object markers, then it can mean both I and me so you should be careful.

eg.
A : 이거 가질 사람?
B : 나

A : Who wants this?
B : I

eg.
A : 그 사람이 사귀는 애가 누구야?
B : 나

A : Who is he dating?
B : Me.