Countercurrent exchange is a process by which a solute is transferred from one solution to another. It is very commonly found in organisms, used for such tasks as absorbing oxygen through fish gills and transferring waste out of human blood.
The process is based on the principle of diffusion--a solute will continue to move from one solution to another until their concentrations are equal. The two solutions flow very close together, but in opposite directions (this is where the name, "counter" current comes from). This is a very clever adaptation that allows almost all of the solute (usually up to 80%) through diffusion, i.e. without spending energy.
It will be easiest to describe how countercurrent exchange functions using an example--fish gills. Fish absorb oxygen from the water into their blood through the gills. A vessel with blood flows alongside a vessel of water, going in opposite directions. At the start of the blood flow, the blood is highly deoxygenated, and at the start of the water flow, the water is highly oxygenated (note that these two ends are not adjacent). The highly oxygenated water moves into the blood nearby, which at this point has become more oxygenated. As the water moves down and loses oxygen, the blood nearby is always has slightly less oxygen, allowing it to be tranferred continuously. At the end, the water has a very small (~20%) concentration, but this is still more than the highly deoxygenated blood next to it.
Human kidneys work in a similar fashion, though they transfer wastes out of the blood and into a waterlike fluid. The process involves several steps however, to allow the kidney to finely control the concentration of the urine produced (in order to conserve water if needed).