US university scholarship for international students
The very top (mostly private) universities will decide which students to admit without considering the financial circumstances of the students or their families.  Then they will look at the student's family's financial situation and cover whatever the student's family is unable to pay.

Slightly less top universities will consider your financial circumstances in making admission decisions, but will admit you (and provide a combination of scholarships and loans sufficient to pay for your education) if you are sufficiently better than their average student.  (How much better is sufficient depends on the financial circumstances of the university, and in particular whether they really need tuition to cover their expenses or cover much of their expenses using interest from prior very large donations.)

The universities that are well off enough to subsidize their international students in this manner almost all do admissions holistically.  That means, for a competitive candidate, someone will take some time to get a complete picture of you as a person from all your application materials and decide on your application on that basis.  They are interested not only in your academic abilities, but also want to consider how much you will contribute to the university community (both academically and socially) as well as how much you are likely to make a contribution to society (and hence to the fame of the university) after your studies.  Note your likely contribution to the university community may depend on who else is already on campus and who else is applying that year.  If you are a good bassoon player, this will help you more in a year when a bassoon player is graduating and only one other bassoon player is applying than in a year when two bassoon players enrolled the previous year and three others are applying.  (It is quite common at top US universities for people to participate in activities unrelated to their studies, so students may continue to play the bassoon in the university orchestra even if they are not interested in further musical studies.)

To make such a judgement, a university will look not only at your standardized test scores, but also at your complete school records, recommendations from your teachers, your activities outside of school, and your responses to the (longer or shorter) essay questions on their application.  If your country regularly sends students to the university in question, someone at the university will be familiar enough with cultural norms in your country to interpret these other parts of your application appropriately.  If not, they will try to learn what they can.

Doing admissions in this way is time consuming and expensive, but if a university is using substantial resources to subsidize its students' education, then it will find it worthwhile to spend some extra resources in trying to get the students it wants.