In most countries, the employer must check that you have the right to work. This has two obvious parts: They have to check that you are who you say you are, and that you have the right to work. Most people are lucky and their identity gives them the right to work. In the USA, that would be any official photo ID that also shows you are a US citizen, typically in the USA a US passport or US driving license, or a document showing that you have the right of work that also has a photo.
In the UK it used to be that any EU passport identified you and gave you the right of work. Nowadays only a UK passport gives you the right of work; an EU passport identifies you and then there is a simple procedure to check that you have the right of work. US citizens would have to show their passport and an additional document showing their right to work.