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Federal income tax is money you pay to the government in Washington on your income. The rate depends on your income. State income tax is the money you pay to the government of the state you live in. Not all states have one (most do) and in those that don't they get you in other ways. The rate depends on the income. City income tax is money you pay to the city (New York City) when you work there (no, you do NOT have to live in NYC to owe NYC income tax).
We pay sales tax - 8 1/4% in NYC on most things (not food). But we don't have VAT like you do in the UK.