저 is for people that are higher than you or for formal situations. 나 is for people that are lower than you or for people that have agreed to let you speak intimately with them. But you would say it like this: 저는 학생이에요 나는 학생이야 Note: Maybe you've seen 저는 학생입니다. This sounds like a formal situation (like a self-introduction in front of a group) and not like a friendly conversation.

저 is used when talking to people who are older than you and/or have a higher status (e.g your boss) 나 is used with your friends of the same age and with younger people. Some possible variants of your sentence are: 저는 학생입니다 저는/나는 학생이에요 나는 학생이야

I think "나는 학생 니다" is informal and "저는 학냉 아애요" is formal, so both sentences are right but they have different level of politeness, but I'm not sure