More specifically do they look at the school you go to?
Minimally. For the most part (at least in STEM) what matters is previous work and/or competition experience.
GPA and current relevant coursework.
Who referred you.
School name, gpa, current relevant coursework, relevant work experience, etc
Basically just think of it like a career opening itself, most places offer internships because it gives them access to cheap/free labor and gives them opportunity to recruit a good employee.