Equity is the value of the home, minus the balance of the mortgage loan. So if you put down 10% when you buy, you start with 10% equity. Your equity would then grow as you pay down the mortgage and as the home value appreciates.

When you build enough equity, you can borrow against it. This is sometimes called a home equity loan or second mortgage. Say you've been paying down your mortgage for a few years, and home prices have increased since you bought so your equity in your home has grown from 10% to 40%. You could borrow 20% of your home's value (home equity lenders typically require 20% equity to remain). A common use is to upgrade or do large capital projects on the home, like paying for a new roof or remodeling the kitchen. But people can also use the money to buy a rental property, fund a new business, buy a car, etc. too.

Not sure what your dad's comment about insurance and equity have to do with each other, so either you misunderstood what he was complaining about or he doesn't understand something.