I added this section because lately we've seen a rash of incorrect statements regarding sensor size, focal length multiplier, and pixel dimensions (from people that know better!).

Sensor Dimensions
Low-end to mid-range digital SLRs use sensors between 24x16 mm, and 18x14 mm, with a focal length multiplier between 1.5 and 2.0.. Professional digital SLRs use sensors that are larger, between 29x19mm and 36x24 mm, with a focal length multiplier of 1.3 to 1.0. At least one professional digital camera has been announced that use a sensor that is larger than 36x24mm.

There is nothing magical about 36x24mm; it happens to have the advantage of all the legacy lenses and accessories, and it also happens to be a convenient size to accommodate pixels large enough to produce the low noise levels required by professionals. There is a conflict between sensor size, pixel dimensions, and resolution (number of pixels) because when you make the pixel dimensions smaller, in order to have greater resolution in the same size sensor, pixel noise goes up (this is a physical law). Despite techniques to reduce pixel noise, the laws of semiconductor physics cannot be violated.

Canon has taken a reasonable approach. In their 16.7 megapixel, high-end professional digital SLR, the EOS 1Ds Mark II, they use a 36x24mm sensor, and a pixel size of  7.2 microns. In their lower-end, 8.2 megapixel, professional digital SLR, the EOS 1D Mark II, they use a 29x19mm sensor, and a pixel pitch of 8.2 microns. These pixel dimensions are large enough to result in very low noise levels, even at higher ISO settings. On the Canon EOS-20D, they use a 22.5x15mm sensor, with a pixel size of 6.4 microns, and still get very low noise.

On the Nikon 12.4 megapixel,  D2X, Nikon chose to use a 24x16mm sensor, with a 5.5 micron pixel size, the smallest pixel size of any digital SLR. It was no surprise when noise levels on the D2X turned out to be higher than on Nikon's consumer level D70, which has 7.8 micron pixels.

The "Smaller is Okay" Propaganda Machine
Do not fall for the misleading propaganda, promulgated by self-interested companies and individuals, that there are no disadvantages to smaller sensors. The expense of digital SLRs with larger sensors, and a focal length multiplier of 1.0 (to 35mm), makes sense for professionals photographers. Just as with the frame size of film, there are technical reasons why the larger the digital sensor area, the better.

Let's examine the two major advantages of larger sensors:

Advantage 1: Signal to Noise Ratio
The larger sensor dimensions permit either larger pixel sizes with the same resolution, or higher resolutions with the same pixel size. The advantage of higher resolution is obvious, but the advantage of larger pixels is less noise at higher ISO settings. It is no accident that the lowest noise, highest resolution, digital SLRs, use full frame sensors.

Even though there have been great advances in active noise reduction (particularly by Canon), you cannot escape the realities of semiconductor physics where noise goes up as pixel size shrinks. A pixel gets its useful information from photons falling on it, the larger the pixel, the more photons per pixel, and the higher the accuracy (the greater the signal to noise ratio). See: Digital Cameras: Does Pixel Size Matter? for a very good explanation of signal to noise ratio and pixel size.

Advantage 2: Focal Length Multiplier
With a unity focal length multiplier, existing lenses, for 35mm film bodies continue to function at their stated focal lengths. This is especially useful at wide-angles, because it eliminates the need for extreme wide-angle lenses that compensate for a smaller sensor. High quality, extreme wide-angle zoom lenses, are extremely expensive (wide-angle lenses are more difficult to manufacture than telephoto lenses). Still, it is possible to compensate for the focal length multiplier by using more expensive lenses, so this advantage is more of a practical and economic advantage.

Why not Go Greater than 36x24mm?
While going to a frame size larger than 36x24mm would permit larger pixels, and yield even greater advantages (and in fact there actually are companies doing this), the huge installed base of 35mm lenses and accessories makes 36x24mm a very practical size, given current pixel dimensions.

Smaller Does Not Always Equal Better
It is ironic that smaller does not always equal better; we've been conditioned to think that everything in electronics gets better when it gets smaller. But for pixel size, smaller does not equal better. Noise on camera sensors is equivalent to leakage current in transistors. As transistors shrink, total power goes down, but leakage power as a percentage of total power goes way up. Leakage translates to noise. Semiconductor companies are struggling to find ways to reduce leakage power as geometries shrink, just as camera sensor companies are struggling to control noise as pixels get smaller.

The conclusion is clear and unambiguous, the smaller the pixel, the less sensitive it is to light, and the more susceptible it is to noise.

Smaller Sensors in Digital SLRs are a Compromise Based on Cost Issues
The 24x16mm, and 18x14mm sensors, used in the non-professional D-SLRs, are not the best choice in terms of image quality; they are necessary compromises for consumers, based on cost issues. It is much more expensive to built a 36x24mm sensor, not just because you get fewer sensors per wafer, but because the larger the sensor, the lower the percentage of good sensors per wafer. This is not to say that the smaller sensor digital cameras are terrible, just that they are not capable of the same quality photos as professional digital cameras with larger sensors.

Watch Out for Long-Winded Diatribes Defending Smaller Sensors
You will occasionally see long-winded diatribes, usually from Nikon aficionados, as to why Nikon has not yet come out with a 36x24mm sensor, despite the pleas from Nikon users. Take these sermons with many grains of salt. Nikon is no doubt feverishly working on a 36x24mm sensor in order to be able to produce a true professional-class digital SLR that will compete against the Canon EOS 1Ds Mark II. The rationalizations in favor of smaller sensors are indeed amusing to read. One of the most amusing, due to the propaganda language, and the faulty logic, can be seen at: http://www.kenrockwell.com/tech/dx.htm.

Larger Sensors are More Desirable, More Costly, and One Company has a Monopoly
The facts support the desirability of 36x24mm sensors, and while they are very costly to manufacture at this time, once yields improve the price will come down. The only reason that the Canon EOS-1Ds Mark II is perpetually sold out, despite its $8000 price tag, is that professional photographers have no alternative if they want to go digital. Once there is some competition in the professional segment, prices will fall; for now Canon can milk the professional market for all its worth. 

Amateur Cameras will Continue to Use Less Expensive, Smaller Sensors
For the time being, amateurs will be using digital SLRs with the less expensive, smaller, sensors. This is a compromise that didn’t need to be made in the days of 35mm film, when everyone bought their sensors in rolls (let’s not forget the failure of smaller film formats such as 110, 126, Disc, and APS, since all but APS failed primarily due to poor photo quality due to the smaller frame size). While there is certainly no reason to go to 18x14mm (the so-called "4:3 standard" used by Olympus), there are plenty of decent cameras available with crop factors of 1.5 to 1.6.

One Advantage to Smaller Sensors
There is one advantage to smaller sensors, used with standard lenses intended for 35mm bodies. This is related to the concept of "shooting center glass," and it also applies to film. The center glass of a lens is optically superior, producing less vignetting. Discarding the outer 20% of the area of the sensor (or film frame) reduces the chromatic aberration. Hence, sensor with a focal length multiplier of about 1.2 would have an advantage, at least in terms of reduced chromatic aberration, over a full frame sensor. 

More Information
For more technical details on pixel size and noise, please read the following articles:

Digital Cameras: Does Pixel Size Matter?
"Larger pixels have reduced noise at all levels, but especially at low signal levels. The obvious improvement still possible would be to reduce the read noise, but that would likely improve large sensors also, thus large sensors will always have an advantage. Whether the difference in noise is great enough for you to choose a larger sensor, and thus likely a larger and heavier camera, is a decision you must make for yourself.

Because good digital cameras are photon noise limited, the larger pixels will always have a higher signal-to-noise unless someone finds a way around the laws of physics, which is highly unlikely."

Digital Photography Essentials #002 "Pixel Size"

The Signal-to-Noise of Digital Camera images and Comparison to Film