In the case of "American", this word is both the name of a person from America ("He is an American") and their nationality (e.g. "He is American" - adjective)
This rule applies if you can make the adjective relating to the nation by adding a final -n e.g India - Indian.
The word "British" is only an adjective describing the nationality. A person from Britain is sometimes called a "Briton" though it is formal and unusual. More commonly, you would say "a British person" for an individual, and "the British" for the entire people.
The rule for British applies to other nationalities ending in -ish e.g. Swedish, and -ch e.g. Dutch, French.