You can say either 'I am American' or 'I am an American'.
If you say 'I am American', the word 'American' is an adjective. Grammatically, it is the same as saying 'I am happy' or 'I am tall'.
If you say 'I am an American', the word 'American' is a noun. Grammatically, it is the same as saying 'I am a teacher' or 'I am a widow'.
Both are correct.
Note that this only generally works with nationalities ending in 'an' and 'i', such as Korean, Brazilian, Australian and Pakistani, Kuwaiti and so on. These words can be both adjectives and nouns.
Other nationalities, such as those ending in 'ish', are ONLY adjectives. So you cannot say 'I'm an English'. You can say 'I'm English', or you can say 'I'm an English woman' or 'I'm an English businessman' or 'I'm an English tourist' and so on.