Each host university has its own application process. You can apply to participate at as many host universities as you are interested in. The application at each host university requires, at a minimum, the submission of an official transcript, a letter of recommendation, and a personal statement or essay.
In the U.S., the application deadline for host universities is in early February, with the exception of the California Institute of Technology, for which the deadline is in mid-February. In Europe, the application deadline for host universities is in early February.
After reading about the Program at that particular host university, follow the link to that university’s summer research program website. You’ll find the Amgen Scholars application there.