What happens is that surface waters, as they get to the vicinity of Europe, grow dense and sink to great depths and begin a slow trip back to the southern hemisphere. When they reach Antarctica, they are caught up in the Antarctic Circumpolar Current, where they are driven onward into the Pacific. The process is very slow—it can take 1,500 years for water to travel from the North Atlantic to the mid-Pacific—but the volumes of heat and water they move are very considerable, and the influence on the climate is enormous.
Thermohaline circulation not only moves heat around, but also helps to stir up nutrients as the currents rise and fall, making greater volumes of the ocean habitable for fish and other marine creatures. Unfortunately, it appears the circulation may also be very sensitive to change. According to computer simulations, even a modest dilution of the ocean's salt content—from increased melting of the Greenland ice sheet, for instance—could disrupt the cycle disastrously.