It's an extremely lengthy process. So, before answering this question, I would like to say that never loose hope while filling the applications. It's a tedious job but worth it.
Firstly, you’ll have to take any standardized tests required for your university. For undergrad, SAT and/or SAT subject tests, for grad studies, GRE or GMAT depending upon the field you wish to go for.
Secondly, if you live in a non-native English speaking countries, you’ll have to take TOEFL/IELTS.
Thirdly, shortlist universities depending on your scores.
Fourthly, apply to your desired universities. Generally, applicants apply in around 6–10 universities. Complete all necessary formalities (documentation, high-school transcripts etc.)
Fifth, wait for admission decisions from one or more universities. Out of the ones that have offered you a place, chose the one you wish to enroll in and accept its offer.
Sixth, complete all financial formalities, reserve a place in dorms/residence halls. Get your student visa and you’re ready to embark on a beautiful journey ahead.
Good luck!