Home equity is the value of a property minus the debts secured against it. On a house appraised at $500,000 with a $300,000 mortgage, the equity is $200,000, and if a second lien of $50,000 is also recorded, the equity is $150,000. That is the whole calculation.
What makes the concept slippery is not the subtraction but the first term. A property does not have one value; it has several, produced by different processes for different purposes, and they routinely differ by tens of thousands of dollars. So a homeowner told they "have $150,000 of equity" has not yet been told anything actionable, because the figure that governs a mortgage insurance cancellation is not the figure that governs how much a lender will advance, and neither is the figure a sale would produce.
Equity is also a residual, not a holding. It is the leftover after two other quantities are set, and only one of them, the debt, responds to anything the owner does. The other moves with the market. That is why equity can fall while the owner makes every payment on time, and why it can be negative. How equity builds in the first place, from the down payment, the amortization of the loan and appreciation, is covered in the guide to real estate.