Let's break it down into parts. There are two words we can work with for 'I': 저 and 나. The uses I will go through are variations of these two. We have two indicators that will be used with these two. They are 가 and 는. 가 is the subject indicator. 는 is the topic indicator. It is my understanding that you use 는 to indicate "this instead of something else" or "This in comparison with something else." Converting your thinking from Spanish/English to Korean takes some time, so this may seem confusing, but it becomes natural after a while. This is how I think of the difference between the two. So now, depending on whether we are using 나 with 는 or 가 the form may change. It is "나는" or "내가." You would never use 나 with 가 and never use 내 with 는. That is just a rule to learn with that. It is either "나는" (I, instead of someone else) or 내가 (I, the subject of the sentence). Using the other form of I, we have a similar rule. It is either 제가, or 저는. Those are always used together, and never interchanged (so never use 제 with 는 nor 저 with 가). 제 is more formal than 나. 난 is a contraction for 나는. So that is the difference. Do you understand all of that?

You use naega 내가 when you are referring to yourself in the sentence, kind of like 'I am'. Nae 내 means 'I', it's used only with the topic marker ga 가. For example, if you say "I am going to the store." you would say Naega gagey ey ga 내가 가게에 가. Na 나 is used in a situation where the speaker is the subject or object of the sentence. It's always used with the subject marker nuun 는 (or object marker reul 를, I'll get to that). If I said, "I don't like football." I would say Na nuun 나는 (usually in colloquial it's contracted to 'Nan 난') chukguga shiro. 난 축구가 싫어. (Sometimes in colloquial Korean the subject '나' is left out of a sentence like this, as well as the topic marker 가 and you could just say 'Nan chukgu shiro or chukgu shiro.) Reul (를) is the object marker, so you would use Nareul 나를 as we use 'me' in a sentence . If you want to say, ''Yuna Kim loves me." you would say Kim Yunaga nareul saranghae 김연아가 나를 사랑해. Just want to explain more so you won't be misunderstood if you are talking to a Korean person ^^ because in Korea you have to be careful in your speech ;D it shows how mannered you are ^^ nae/na 내/나 are used in very informal speech (ban mal 반말). As a general rule, be careful when using ban mal and all forms of the word 'you' (ney,ni,no,dangshin 네,니,너,당신). It's only ok if you are talking to children, close friends of the same age or younger, family (but never older people) or your lover.

I think that's one of quite interesting parts of the Korean langauage, and you can find it in the Japanese language as well. (but not exactly same ^_____^) Here is a thing why there are lots of versions of "I am". When it comes to "I am", we count several things before we speak it out. 1) The relationship with listners.(we have got 7 speech levels) (Please visit this link "http://en.wikipedia.org/wiki/Korean_honorifics" especially "speech levels" at the bottom of the article) 2) The situations where you are in.(formal, informal) 3) Your position in the workplace. Looking at your question, 1) Naega (내가) is not polite and it's used among friends. the polite one is "제가(Je ga)" which is used in formal situations or to older persons. 2) Naneun (나는) is almost the same as "Naega" and used in the same situations where "내가" is used, but as Steve said the main difference is "markers" or "indicators" in it. 3) Nan (난) is the short version of "나는" The polite one is "전(Jeon)" which is the short form of "저는(Jeo neun)" so before you speak or write something in Korean, you should think of the above things like relationship or situations. FYI, *Markers(or indicators) make a noun a subject or an object in a sentence. For example, A NOUN could be a subject, object or topic in a sentence before putting a specific marker in it. Putting subject makers like 이/가 it can be a subject. Likewise you can put object markers(을/를) to make the noun an object. ^____^