The word American is used to mean a person or a thing from the United States or any country in the Americas - North America, Central America, and South America. In English, the most common use of this word is to mean a person or a thing from the United States.
Even though the islands in the Caribbean Sea are close to the Americas, people who speak English do not usually use the word "American" for people or things from these islands.
Due to the ambiguity of the word, some languages avoid using the word to refer to someone and/or thing from the United States and use more specific demonyms instead, such as Spanish estadounidense, Esperanto usonano and German US-Amerikaner, for example. Yankee is sometimes used within the English language too (though the word historically refers specifically to an inhabitant of New England). Usonian (from Usonia) is a alternative demonym to "American" proposed by architect Frank Lloyd Wright.
A Native American is someone who is mostly descended from the people who lived in the Americas before the Europeans arrived. Native Americans are also called First Nations and Indians.
There is no language "American." Some important languages used in the Americas are English, French, Portuguese, and Spanish. Some people still use Native American languages. Quechua is the largest.