It's been a while since I went through this but most of this is still relevant.
The application process (though mostly online) is quite long and needs to be planned at least a year in advance.
If you go to most admissions websites, you will see a few things that most American Universities require:
SAT and SAT Subject Tests (usually 2) - SAT is a regular standardized test with English and Maths sections. Doing well on these require some serious prep and most people do it a year before they apply.
Application Essays (Besides the Statement of Purpose) - There are about 5 additional questions usually about 300 to 500 words each and the questions can vary considerably depending on the University. These essays are a way to show them how well-rounded you are and it helps if your child is involved in a number of things. Writing these can take a lot out of you.
Letters of recommendations (usually 2 or 3) - 2 from teachers and 1 ideally from somewhere you've worked (It's quite common for students in the US to do some sort of internships/part-time jobs while they're still in school)
Transcripts (Summary of the classes you've taken along with grades and how many credits these equate to). They usually want your grade history for the the High School years (Y10 to Y13). Can take a few weeks to get these and not all schools have this in the correct format that American Universities expect.
For Ivys or MIT/Stanford/Cal Tech, you would need to prove to them that you are REALLY good at your field (whatever that field is). To do this, you need to do something ideally at a national level. MIT for example pretty much expects their International students to be part of their country's Olympiad team. Brown/Dartmouth/Cornell are relatively easier to get into (but still difficult).
If you are applying for Financial Aid then you need to provide a bunch of documents. It's probably similar to applying for a bursary at an independent school. Just want to add that not all Universities are 'Needs Blind' and you are less likely to get into some places if you ask for financial aid.
Lastly, once you get accepted, you need to apply for a student visa and the US government requires proof of funds for the entire first year fees (even if you are getting financial aid). This means getting a letter from your bank saying you have $xx,xxx in your account.