What is equity? Why is it important as used in this quote?

Equity is the net ownership you have in an asset. In this context, it is the difference between the value of the real estate asset and the debt owed on it. So, for example, if you bought a home for $400,000 and took a mortgage loan of $375,000, you have equity of $25,000. If the value of that home drops to $350,000, you actually have a negative equity. At that point you would simply allow the lender to foreclose on the property rather than repaying the $375,000.