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.

Both are correct. One can say "I am American." And one can also say, "I am an American." "I am an English" however, is incorrect. At least in the American dialect, that sentence is not correct. Perhaps a native speaker of British English can say whether it is correct in their dialect or not.

Some nationalities can function as both an adjective and a noun. 'American' is one of them. Therefore you can say 'I am American' where 'American' is an adjective or 'I am an American' where 'American' is a noun. 'English', when referring to nationality, can only be an adjective. Thus you can only say 'I am English', NOT 'I am an English.' There is no easy way to work out which nationalities function as both adjectives and nouns, and which are just adjectives. You simply have to learn them.

They are both correct. Note that for the English, one would say "I am English," but NOT "I am an English."

Similarly, a common error in China is "I am a Chinese", which is incorrect. "Chinese" can only be used as an adjective. The correct thing to say is "I an Chinese."