Private mortgage insurance (PMI) covers the lender against borrower default. If you borrow more than 80% of the value of a home when you refinance you will be required to pay PMI. If you don't have 20% available for a down payment, consider taking a first mortgage and a second mortgage.
Some people call this strategy an 80-10-10 loan: 80% on the first mortgage, 10% on the second mortgage, and 10% down. (Of course this strategy assumes you have sufficient funds available to put 10% down.) While the second mortgage typically has a higher interest rate than the first mortgage, the net result is a lower payment in total if PMI is eliminated. And, mortgage interest is likely to be tax-deductible while PMI payments are not, which further increases your savings.