A person, group or business entity purchases enough shares of your company's stock to enable a hostile takeover or at least to have a say in your operations and management. They offer to let your company buy back the shares, but at a much higher price. In exchange, they won't continue with the attempted takeover and won't purchase any more of your stock. That's called "greenmail."
Laws and regulations preventing companies from repurchasing shares of their stock at inflated prices have helped thwart attempts at greenmail, which saw a rise in the 1980s when corporate takeovers were at a high. In 1987, for example, the Internal Revenue Service (IRS) started taxing profits from greenmail at 50%.