The term "need blind".

Q & A for need blind universities in US

Q: Does the amount of money I request for from the university affect my chances of getting in?
A: Yes if it is not one of the 7 colleges I mentioned in the last post, no if it is. Some lower-tiered universities however will be glad to give you $$ for you to attend their university. It's up to you if you want to take up their offer. I know of a friend going to the University of Dayton on a free ride.

Q: So should I apply for financial aid? How much does it lower my chances by?
A: Lowers it DRASTICALLY. Stanford's overall admission rate is ~10%, which is already very low. For internationals who apply for financial aid, it's rumoured to be ~3%, probably even less if you want FULL AID.

Q: But I really have no $$ to go.
A: You have 2 options. The first is that you apply for financial aid and lower your chances drastically, the 2nd is that you apply normally without requesting for financial aid and hope that you can get your $$ somewhere else.

Q: Aha! I know! Can I tell them that I want to be considered for both financial aid and no financial aid? Then if I am not good enough in the FA pool I can be considered for the non-FA pool and I will have the best of both worlds!
A: No. You either say you have $$ or no $$. Period.

Q: Hm, how about I get in first, then I tell them I need $$ and then they will have to give it to me!
A: Doesnt work either, the application form clearly states that once you say you dont need $$ you are not going to receive any from the university in the next 4 years.

Q: So which internationals apply for FA?
A: Internationals who know that they definitely definitely definitely cant go to the university even if they get in because they have no $. Or any international who is applying to a need-blind college.

Q: What if I want to apply for financial aid?
A: Then you fill up some forms about financial background and amount of $$ you need. Take a look at the Harvard website for more info. http://www.admissions.college.harvard.edu/apply/international/international_aid.html
And this form: http://isites.harvard.edu/fs/docs/icb.topic551531.files/freshman_foreign_aid_app.pdf

Q: I just want to apply as a regular applicant without financial aid.
A: Then you dont have to care about the FA forms. Those are only for applicants who need FA and those have to be sent separately to the respective colleges.

Q: How much $ can I expect to get?
A: It depends on each school, but a general indicator is the annual family income that you have. You can try using this financial aid calculator for a rough gauge: http://www.finaid.org/calculators/finaidestimate.phtml
This website is about the rough number of people who get $ from Harvard: http://isites.harvard.edu/icb/icb.do?keyword=k51861&pageid=icb.page246751
This is another website that may help in budgeting: http://www.edupass.org/finaid/costs.phtml