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Equity is the financial word for ownership: to have equity in something is to have financial ownership of it.
In the real estate markets, equity is the difference between the value of the property and the owner's debt against the property (i.e., the mortgage or property loan) - though the title of ownership may be in her name, the owner's financial ownership (and perhaps "actual" ownership, depending how you look at it) is only as great as the equity she holds in the property. In the securities market, equity refers to ownership in companies.
Security refers to an investment of money in the form of a contract, which itself has a monetary value and can be traded in a marketplace.
An equity security represents financial ownership in a publicly traded company. Equity securities are most commonly issued, purchased and traded in the form of common stock and preferred stock. Each share of stock represents a portion of ownership in the issuing company, and is therefor called equity.
Other types of equity securities are warrants, which allow the holder to purchase stock from the issuer at a certain price over a long period of time, American Depository Receipts (ADRs), which represent the receipt of US company shares in a non-US corporation, and Real Estate Investment Trusts (REITs), which are publicly traded companies that manage real estate investments with the intention of earning profit for shareholders.