She is Korean is the correct way to talk about her nationality. She is a Korean- we are missing some sort of noun at the end. For example- She is a Korean girl, she is a Korean actress.

She is Korean is correct but after she is a Korean, you should say something. For example: she is a Korean student.

As GuideDogSaint said there is no rule about this. Bit there are some nationalities where there is a single word (like a Canadian) and others where there is not (a Korean *person*). To me using the Asian ones like "a Chinese", "a Japanese" or "a Korean" sounds rude. Others, like "a French" or "a Spanish" don't sound rude, they just sound wrong. I'm not sure why.

This is a good question. For some nationalities there are clear differences between the adjective and the noun. He is Spanish. He is a Spaniard. She is Icelandic. She is an Icelander. Others use man/women He is French. He is a Frenchman. She is English. She is an Englishwoman. For some they are the same He is American. He is an American She is Canadian. She is a Canadian. Others don't really have an adj. He is from New Zealand. He is a New Zealander. I think 'She is Korean', 'She is a Korean' are both fine and mean the same as an answer to the question 'Where is she from?' Not sure there is any general rule with this.