Thermal-Electric Cooling (TEC): 

TEC cooling is starting to become a little more common place. This method is sometimes referred to as Peltier cooling. It's best described as a solid state heat pump. Without going into the science of how it works, it uses an electrical current to make one side of a conductive plate hot, while the other side is made very cold. This allows a processor to be chilled to below ambient temperature. The hot side is usually cooled using either conventional air or water methods. It's a cheap way of doing this but consumes quite a bit of power in order to be effective. It also doesn't completely eliminate the need for water or air. Some commercial coolers such as the Monsoon II use TEC cooling combined with air but their performance is mixed.

Phase Change Cooling:

Most people are familiar with phase change cooling. Your refrigerator uses this method, as well as most air conditioners.  It works by using a compressor to compress a gas to a liquid, making it hot. A heat exchanger removes the heat from the liquid. The liquid expands and is converted to a super cooled, high pressure gas. The cold gas cycles through tubes to your processor where it removes the heat. It is then cycled back to the compressor and exchanger to be converted back into a liquid and dump the heat. This is perhaps the ultimate cooler you can buy off a store shelf, though they are very uncommon. There are a few reasons why nobody uses these. First of all, they're expensive to both buy and operate. I only personally know of one place that sells these and the cheapest one I could find was $800. You can buy an entire full size fridge for that. A high end, custom water cooling system will only cost half. The cheapest one I found also consumes 440w of electricity, which is as much as an entire typical high end computer does. The second downside is that the units are exceedingly large. They're about half the height of a mid-tower case. They also require you to drill a hole in the case to fit the hose inside. Lian-Li does make pre-modded cases for these coolers though. Lastly, these systems are typically designed to cool the processor only. An air or water system will still be required to cool other components such as the GPU.  Therefore, while they offer the best off-the-shelf cooling performance of any other method, they are impractical and unnecessary. Most modern processors, even ones heavily overclocked, do not need this much cooling. 

Dry Ice and Liquid Nitrogen:

The most exotic cooling method I've seen is the use of liquid nitrogen or dry ice. This is the ultimate sub-zero cooling method. You can't buy parts for these. Typically, people using these make custom, home-made setups. Liquid nitrogen is exactly that, nitrogen that has been cooled so much it becomes a liquid. Dry ice is carbon dioxide cooled enough to become solid. Dry ice is warmer than liquid nitrogen but both are very cold. They offer the ultimate in cooling but are also clumsy, dangerous, and hugely impractical. These cannot be used for everyday use. Typically, these methods are only used in overclocking competitions that push processors beyond their limits. 

The Problem of Condensation:

Aside from the problems I've mentioned, all sub-zero methods of computer cooling present a single challenge. Moisture. When objects are cooled below ambient, condensation forms. Since water conducts electricity, condensation can fry your components. Therefore, the processor based has to be sealed with silicone in order to prevent moisture from touching electrical contacts. This additional step is not necessary with air or water.

Oil Cooling:

See the electrical transformer pots on your street. These are cooled using oil,  a passive cooling method. Oil, like water, can hold and release more heat than air can. A couple of years ago, Tom's Hardware displayed a computer cooled with cooking oil. Oil is a non conductive substance so it was possible to submerge all components in a sealed case full of off the shelf cooking oil. The system has the advantage of running in near complete silence. This method presents a few obvious problems. First of all, it's dirty. You will need to periodically change the oil and you will likely never get all the grease off your components. Secondly, the entire case must be water tight, with all connections and wires sealed with silicone. This has not deterred some users. Some people claim more success with using motor oil. As with several other exotic methods, oil cooling is impractical. It also requires the processor bases to be sealed as with the sub-zero methods and it still requires heat sinks. Like air and water, it is an ambient cooling method, meaning the computer cannot possible get cooler than ambient air temperature. Therefore, it's just easier to use air or water. 

Passive Cooling: 

Passive air cooling isn't really an exotic method but it is not as common as active air and water. However, it has seen a resurgence in interest from the home theater PC crowd looking for systems to be absolutely silent. How it works is simple. It uses very large heatsinks with more surface area to dispurse heat to the air without using fans. Apple uses a passive hybrid system in some of their computers. Hybrid passive systems (which is a name I made up) have a fan that only kicks in if the component being cooled reaches a certain temperature. The fan shuts off once a safe temperature is reached. Therefore, they are quiet most of the time. These systems are typically used for processors with lower clocks and overclocking with them is not really possible. However, they are a must when silence is golden. Passive water systems also exist, such as Zalman's Reserator series. These systems use very large fanless radiators but still have a pump. There was also a German company making these but the name escapes me.

General Cooling Tips:

I thought I'd end up with some general cooling tips to improve your computer's cooling performance. First of all, keep the system tidy. In the case of air cooling, tuck any necessary internal cables out of the way so airflow is unobstructed. Make sure to remove any dust from radiators or heat sinks regularly using compressed gas tins and not a vacuum. Vacuums can build up static electricity. Dust acts as a surprisingly good insulator for heat so getting rid of it is important. For water, check the system on a monthly basis for clogs and corrosion and change the coolant periodically. Flow meters can be bought to monitor performance. 

The place where you keep your computer is also important. You want to keep the system in a cool, dry place away from heat sources. These include direct sun, heat vents, household radiators, portable heaters, etc. Ideal room temperature should not get above 25c (77f) for optimum cooling. You should invest in at least a window air conditioner for your computer room for the summer months, or you can keep it in a cool basement off the floor.