There are two main methods by which image noise can be reduced. The most common and effective method is by using smoothing filters during image processing. There are several different types of smoothing filter, but the one most commonly used in image processing is called a Median filter. This works by comparing each pixels to the ones surrounding it, and if is has a brightness value that is radically different from its neighbours, then it is replaced by an artificial pixel with the average value of the neighbouring pixels. This eliminates noise effects, but also has the effect of reducing detail and contrast. Compare these next two images. The first is unfiltered, the second has had a median filter applied.
As you can see the second image is much "cleaner", but details such as the pattern of the bricks has been reduced. This is a fairly crude example to illustrate the point, but image processing similar to this is carried out routinely by your digital camera to reduce image noise, which is one of the reasons that noise-reduced high-ISO shots have less fine detail than low-ISO shots. Different manufacturers use their own proprietary processing systems, and some cope better than others, which is why I always include ISO noise test shots in my camera reviews.
The other main type of noise reduction is a process called "pixel binning", although this is less widely used in recent digital cameras. In this process, the signals from groups of four, nine or even 16 adjacent photocells are grouped together into a "super pixel". This has the effect of increasing the signal to noise ratio, but of course it also reduces the effective resolution of the image. It is effective at reducing noise, but it also reduces the quality of the image.
In the end, there is really only one effective remedy to image noise, and that is to use larger, more sensitive photocells. Digital SLRs have a major advantage in this area, since they have physically larger sensors, but as megapixel-counts continue to rise then even Four-Thirds and APS-C sensors may begin to suffer from increased noise problems. As for compact camera sensors, technology continues to improve, both in image processing and sensor design. Fujifilm has gone some way in the right direction with its Super CCD technology, and many models in the FinePix F-series have superior high-ISO performance as a result. Sony's latest SuperHAD sensor also has larger photocells and can produce good results at moderately high ISO settings. We will undoubtedly see further advances in the future, but for now image noise is something we just have to accept.