What an extra payment actually buys
Every pound or dollar added to a mortgage payment goes straight against the balance. It does not sit anywhere, it does not reduce next month's payment, and none of it is interest. It simply makes the balance smaller than the schedule expected, which means every month after it carries less interest than it would have.
That is why the saving compounds and why it is so much larger than the money put in. The effect is strongest early, when the balance is high and most of the payment is interest anyway, and it fades towards the end when there is little interest left to avoid.
Two ways to ask the same question
Most people arrive with one of two questions, and they are the same arithmetic read from opposite ends. Either you know what you can spare each month and want to know what it buys, or you know when you want to be finished and want to know what that costs.
This page answers both. Working back from a date, the figure it gives is the smallest extra payment that gets there. Because payments move in whole cents while the finish line moves in whole months, the smallest payment that reaches your date will sometimes clear the loan a little sooner than asked. The page says so and names the month it actually reaches, rather than repeating the date you typed back at you.
The rule that catches people out
Paying extra does not bring forward the day mortgage insurance stops. This is the opposite of what the balance suggests and the opposite of what most people assume, and in the United States it is written into the statute: the automatic termination date is fixed to the original amortization schedule, whatever the balance happens to be on the day.
There is a separate route, which is asking the lender to cancel once the balance is genuinely low enough. That one is a request rather than an automatic right, and it usually comes with conditions about payment history and a current valuation. The distinction is worth knowing before counting an insurance saving that will not arrive on its own.
Before paying down a mortgage at all
A mortgage is usually the cheapest money a household will ever borrow. Any debt at a higher rate, and a credit card almost always qualifies, is worth clearing first, and the debt payoff page is built for exactly that comparison. An emergency fund generally comes before both, because money put into a house is hard to get back out in the month you need it.
There is also the question of what the same money would earn elsewhere. Paying down a loan is a guaranteed return equal to its rate, which is a genuinely good return when rates are high and a poor one when they are low.