Financial Aid: The college wants to provide you aid. Aid means you need help and the college recognizes that and would like to help you financially. They determine that you need it by looking at your family's income tax returns after you get admission into the college. How much aid you get depends on your family's income tax return. Some colleges like MIT and Harvard profile financial aid and are need-blind, i.e. your chances of getting admission is not effected by whether or not you need financial aid.
Financial Scholarship: Merit based financial rewards, i.e. if you apply for financial scholarship, your achievements need to be stronger than the applicants who don't need financial scholarship. There are two different piles the college maintains, one for people who need the scholarship and once for people who don't. Clearly, to stand out in the first pile your merits and achievements need to be better than to stand out in the second pile.
Financial Sponsorship: A loan that you take from the government or form the bank in-order to sponsor your education, which you pay back after you graduate. Usually in monthly installments starting 6-months of 1-year after you graduate, but timeline may vary. Often sponsorship will be restricted and only available to students belonging to the same state / country as the university is in, so be careful to check if you're eligible.
I've seen a lot of people be confused about the above three terms. Sometimes the colleges also mean one thing but might use the wrong term. Make sure you understand what each of the above are and how they differ, and which ones the college you are applying to offers.