Refinancing basically means that you take out a new loan (aka: mortgage) at rates and terms different than the original financing.
Here's the backstory:
Most people don't have thousands and thousands of dollars sitting around that they can use to buy a house. So, they take a loan, called a "mortgage," from the bank. That's also called "financing" a house.
The mortgage has an interest rate and terms. The interest rate is how much it will cost to borrow the money and the terms are how many years the borrower has to pay back the loan.
Rates and terms change over time, some times for the better (cheaper to borrow money), sometimes for the worse (more expensive to borrow money).
Refinancing means that you take out a new loan (aka: mortgage) at rates and terms different than the original financing.
For example, say your parents "financed" a house in 1985. They borrowed $100,000 at 20% (that's the interest rate) for 40 years (that's the term).
Now that it's 2018, interest rates are lower - we'll say 8%. That means they can pay 12% less on the money they borrow. (20% - 8% = 12%).
So they "re-finance" the house: they borrow $20,000 at 8% for the remaining 7 years of the loan and save 12% on that $20,000, which is roughly 20,000 X 0.12 = $2,400! Had they not refinanced, they'd have paid 20,000 X 0.20 = $4,000 extra!