Both are correct ,but they are used in different occasions or as an answer to different kind of questions. 'I am American' When someone is inquiring about your nationality. 'What is your nationality?' 'I am American' 'Are you British?' 'No, I am American.' When you state it as a fact as part of your self-introduction referring to yourself as an American among other Americans you use 'an American'. 'I am an American living in France.' 'I am an American working overseas for a multinational company.'

Both "I am American" and "I am an American" are correct. But "I am a/an ______ " is not always correct or appropriate for other nationalities. For example, "I am a French" is just wrong. You have to say, "I am French." "I am a Chinese" is also NOT good. For Americans my age, it makes us really uncomfortable to hear that. We were taught never to say, "He is a Chinese," as it is considered rude or politically incorrect, maybe even racist. It is much better just to say, "I am Chinese." So please just use "I am American," "He is French," "She is Chinese" and don't worry about the other way of saying it.

Both are correct, though I think most would just say "I'm American". The first one sounds better if you add something; "I'm an American citizen/man/etc.."

Both are correct.

I am an American is correct