I'm assuming you're not Canadian or Mexican. Some US colleges have special provisions making it easier for Canadians and Mexicans to obtain aid than other international students. I'm also assuming that your family does not make more than the equivalent of ~$100k USD. It's an arbitrary number, but for many making that much, financial aid does not really help too much.
There are two routes that you can follow, depending on what colleges you qualify for. You can either attend a very prestigious college that provides need-based financial aid, or you can attend a college that gives out merit scholarships (some very prestigious colleges, like Rice, Carleton, and Swarthmore, have specific merit scholarships for international students, but it's very limited).
I'll concentrate more on the first route, as I'm more familiar with it and as it's more transparent. However, to go either route, you'll need some high test scores and GPA. Aim for 2000+ SAT or 29+ ACT (and no…an 800 in math and a 600 both in CR and W does not look very good), hopefully higher. American colleges, many of which are cash-strapped, are reluctant to give money to those without exceptional aptitude or skills.
The best of the best
These colleges are need-blind (how much your parents make won't impact admission decisions) and meet 100% of aid for international students. These colleges are considered among the best in the world and are immensely competitive, especially for international applicants.
- Amherst College (MA)
- Harvard (MA)
- MIT (MA)
- Dartmouth College (NH)
- Princeton (NJ)
- Yale (CT)
In the likely chance that you don't get into one of those…there's the still-very-competitive-but-not-as-much-as-the-6-above
Okay, Dartmouth and Amherst aren't as competitive as some other schools on this next list, at least in normal US-applicant-heavy rates. These colleges do look at how much money your parents make and have, and it can be factored into the admission decision. However, of those they accept, they promise to meet 100% of need. While it sounds like they use this as an escape hatch to avoid accepting poor international students, it's not…completely. Richer international students will have an easier time getting in, but they do all have a number of poor international students. These are all still very good colleges, despite that some of the liberal arts colleges don't have the highest name recognition.
- Barnard College (NY)
- Bates College (ME)
- Bowdoin College (ME)
- Brown University (RI)
- California Institute of Technology (CA)
- Colgate University (NY)
- Columbia University (NY)
- Hamilton College (NY)
- Haverford College (PA)
- Middlebury College (VT)
- Oberlin College (OH)
- Occidental College (CA)
- Pomona College (CA)
- Smith College (MA, GIRLS ONLY)
- Swarthmore College (PA)
- University of Chicago (IL)
- University of Pennsylvania (PA)
- University of Richmond (VA)
- Vassar College (NY)
- Wellesley College (MA, GIRLS ONLY)
- Williams College (MA)
Basically most other top colleges are some combination of need-aware and don't necessarily meet 100% of aid.
International students admitted to Stanford, Colby, etc. are not guaranteed that all their need be met, even though finances were a part of the admission decision. But the colleges will oftentimes attempt to get aid to these students. Again, competition for this money is tough. But it comes through and works out for many students.
But some top colleges are reputed to be bad at delivering need-based international aid
Including Cornell. I could guesstimate some other colleges (probably top colleges with relatively smaller endowment:student ratios), but I don't want to give away false info.
Merit Scholarships are a viable option
The college (which receives state money) I'm going to doesn't offer merit aid to international students, and this goes for many American universities. The colleges that do offer merit aid may not necessarily be ranked higher than the university in your home country, and aren't too lucrative for international students because a. home country's cost is cheaper, even with merit, and b. little name recognition in home country. Now, there's no way I can list every American institution willing to give merit money to international students. I've read that the University of Alabama awards its out-of-state scholarships to international students (and they're guaranteed scholarships, with full tuition for 32+ACT or 1400+SAT (CR and Math)). Look up some colleges ranked outside of the top 30 and call them up or email them if their websites describing international aid are inadequate (they usually are). Guaranteed scholarships are your best chances because they're 100% likely. They're just not the best aid potentially available if your family's income is very low.