Chilling wort without a chiller
While not perhaps technically a type of equipment, most homebrewers start out cooling the wort by using an ice bath. For a partial boil, the only real equipment you need is a sink or tub filled with a mixture of ice and water. Some brewers also use sanitized bottles filled with ice, or even dry ice, to attempt a quick cooling without additional equipment or water use.
However, for a full wort boil, or cooling any liquid volume of five gallons or more, an ice bath is impractical; instead, you will need a dedicated piece of equipment.
 Types of wort chiller
Wort chillers come in several types, all of which work on the same basic principle: cooling a metal surface by flowing cold water past it, then exposing the wort to the cold surface.
The colder the input water, the more potential it has to pull heat out of the wort. In some cases, tap water is not cold enough to cool the wort fast enough to be practical. In this case, the brewer has two options. Tap water may be fed through a pre-chiller before being fed through the chiller; the most common type of pre-chiller is essentially an immersion wort chiller immersed in an ice bath. Another option is to pump icewater directly through the chiller (using an aquarium pump if necessary) rather than tap water.
 Immersion chillers
Most homebrewers start off with a simple immersion wort chiller. An immersion chiller is simply a coil of metal, usually copper, through which cold water flows. The coil itself is then immersed completely in the hot wort.
One benefit of the immersion chiller is that it is easier to sanitize than many other types, since the only part of the chiller that touches the wort is the outside of the coils, which can be simply and easily sanitized or even sterilized by immersing the chiller in the boil during the last few minutes. One drawback is that under certain circumstances, copper can develop contaminants that can be transferred into the wort.
All else being equal, a larger diameter and/or longer length coil will cool faster. Both dimensions contribute to the overall surface area between cold water and hot wort. They also contribute to the volume of cold water exposed to wort at any given moment. Larger diameters may only benefit to a point as the very center of water the tubing may remain colder (unexposed). Longer lengths benefit only to the point where the water is heated to near equilibrium with the wort temperature. Conversly, if your output water is more than 10-20 degrees lower than your wort temperature, you could benefit from a longer length coil. In general, 3/8" OD x 25' long coils are well suited for 5 gallon batches while 3/8 or 1/2" OD x 50' coils are better suited for 10 gallons batches.
While cooling, wort does move via convection through the volume of the boiling kettle. This movement introduces hot wort to the surface of the chiller while cooled wort moves out of the way. However, this natural process can be helped along by swirling or whirlpooling the column of wort by a gentle stirring with a sanitized spoon. Recognizing the importance of keeping the wort moving, some brewers go as far as using pumps or motorized stiring devices to reduce manual intervention.
 Counterflow chillers
The Counterflow wort chiller or "CFC" is a standalone heat exchanger built as coil within a coil. Unlike an immersion chiller, hot wort flows by gravity or is pumped through the inner coil (usually made of copper). While the wort flows, cold tap water is fed into an outer jacket that surrounds the copper coil. This outer jacket is rubber or vinyl hose in budget CFC's but may also be made of larger diameter copper. The counterflow gets its name from the fact that the cold water enters at the end of the coil in which the wort exits (they each flow in opposite directions) presumably to maximize the heat differential. CFC's are designed so that the wort will exit the chiller at pitching temps making it capable of cooling an unlimited batch size. The potential disadvantages to using a CFC are the requirement for cleaning/sanitizing the inside of a long copper coil and also a way to get your hot wort into the coil (bulkhead or metal racking cane).
Counterflow Chillers can be easily constructed using 3/8" OD copper and 5/8" ID high temp garden hose following the directions in DIY:How to make a Counter Flow Chiller.
 Plate chillers
A plate wort chiller can be thought of as a compact version of a counterflow chiller. Like an immersion chiller, the plate chiller is a standalone heat exchanger where both hot wort and cold water are brought together outside of the brew pot. However, in the case of a plate chiller, the actual heat exchange takes place across a series of metal plates rather than in a single hose or tube.
Plate chillers are said to have the best efficiency in water usage and obviously take up a very small space which may be a concern in a compact brewing system. One drawback is that the tiny passages and inability to dissassemble makes cleaning difficult. Care must be taken to strain the pellet hop particles out of the wort prior to running through the chiller by using a hop stopper or similar straining device.