In order to apply for an internship in the United States, what exactly will I need to send? The academic transcripts of my grades? The TOEFL? The GRE?
There is absolutely no "recipe" that anyone can give you, since every company will have its own expectations. Some may want just a résumé and a list of references; others may want your transcripts, a detailed CV, cover letter, and letters of reference; and still others may want something else instead.

You need to consult with the individual company's HR information.

(If you're talking about some sort of internship at a university, the same general rules apply.)