It is the value of the home minus how much you owe on the home. When you buy it, your equity in the home is essentially your down payment. You increase equity by two means:

#1) Pay down your mortgage

#2) The value of the home increases

You can actually have negative equity if you purchase a home at the top of the market. When the market falls you may owe more on your home than it is worth (happened a lot around 2008).

Insurance has nothing to do with equity, so you or your father misunderstood what was going on here. I do not believe a mortgage lender can require PMI (private mortgage insurance) after escrow closes and your equity falls below 20%. It is possible if the home value went down and your father tried to refinance they would require PMI if his equity dropped below 20%.