If you only have enough available for a minimum down payment, your choices of loan program will be limited to only a few types of mortgages. If someone is giving you a gift for all or part of the down payment, your options are also limited. If you have enough for the down payment, but need the lender or seller to cover all or part of your closing costs, this further limits your options. If you borrow all or a portion of the down payment from your 401K or retirement plan, different loan programs have different rules on how you qualify.
Of course, if you have enough for a large down payment, then you have lots of choices.
Your loan choices include such varied programs as conventional fixed rate loans, adjustable rate mortgages, buydowns, VA, FHA, graduated payment mortgages and all the varieties of each.