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.
(As for the question of how anyone could possibly figure out how long it takes a drop of water to get from one ocean to another, the answer is that scientists can measure compounds in the water like chlorofluorocarbons and work out how long it has been since they were last in the air. By comparing a lot of measurements from different depths and locations they can reasonably chart the water's movement.)
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.
The seas do one other great favor for us. They soak up tremendous volumes of carbon and provide a means for it to be safely locked away.