AKA a drogue, storm drogue, drift sock, storm anchor, drift anchor, para-anchor, or boat brake.
A sea anchor is any device used to increase drag and that is trailed along behind a boat. These most often take the form of weighted 'parachutes' made of canvas. Smaller cone-shaped ones are more often referred to as drogues. Other sea anchors take the form of various wooden or metal frames covered in canvas, or improvised forms such as a rope with bits of canvas tied along it or a light anchor wrapped in canvas. The U.S. Coast Guard has warned that common improvised sea anchors made from tires or long lengths of chain are not effective, but these are still in use in some areas.
There are many reasons to use a sea anchor. They will slow the boat, meaning that it is less likely to go too far off-course during a storm or at night when the crew is asleep. When tethered in a 'Y' of ropes to the bow and stern of the boat (i.e. using a bridle), adjusting the bow and stern ropes can effectively maintain the boat's heading relative to the wind, and thus into the waves; this prevents a boat from turning parallel to the wave and capsizing. In some storms, the waves are large and steep enough that simply slowing your descent down the wave can avert disaster.
The same 'Y' rigging setup can allow a sailboat to steer without a rudder, although in this case you will want to use a smaller drogue. Drogues can also be used when sailing in heavy wind or gusty wind to allow for better control over the speed of the boat.
Sea anchors may also be used when a boat is being towed to prevent lateral drift. Fishing boats will use sea anchors to minimize drift due to wind -- they will, however, increase drift with the current. Finally, there are some situations where a sea anchor is an effective way of travelling against the wind, if the current is with you. However, this is hardly a common use of the anchor.
The most effective use of sea anchors can be quite complicated. Different sizes and various lengths of line are used for different applications and situations, and due to considerations of weight and on-board space you will find many different solutions to various needs. For example, the length of the line is important not only in that a longer line provides more shock absorption (for when the boat changes speed as it encounters waves and swells), but also that its length be in phase with the waves (in calm seas, where a boat travelling down a swell can increase the load on the line dramatically if the anchor is not doing the same in an adjoining swell) or out-of-phase (in stormy weather, where the boat and the anchor might both the rolled by adjacent waves). On the other hand, boats that spend little time at sea may want a drogue on a short line, and help maximize shock absorption with a small anchor midway along the line, providing a curve that will shorten the line when load is low and allow the curve to flatten when the load is high.