A31a: In computer graphics circles, "aliasing" refers to the phenomenon of a high frequency in a continuous signal masquerading as a lower frequency in the sampled output of the continuous signal. This is a consequence of the discrete sampling used by the computer.
Put another way, it is the appearance of "chuckiness" in an still image. Because of the finite resolution of a computer screen, a single pixel has an associate width, whereas in mathematics each point is infintesimely small, with no width. So a single pixel on the screen actually visually represents an infinite number of mathematical points, each of which may have a different correct visual representation.
Q31b: What does aliasing have to do with fractals?
A31b: Fractals, are very strange objects indeed. Because they have an infinite amount of arbitrarily small detail embedded inside them, they have an infinite number of frequencies in the images. When we use a program to compute an image of a fractal, each pixel in the image is actually a sample of the fractal. Because the fractal itself has arbitrarily high frequencies inside it, we can never sample high enough to reveal the "true" nature of the fractal. Every fractal ever computed has aliasing in it. (A special kind of aliasing is called "Moire' patterns" and are often visible in fractals as well.)
Q31c: How Do I "Anti-Alias" Fractals?
A31c: We can't eliminate aliasing entirely from a fractal but we can use some tricks to reduce the aliasing present in the fractal. This is what is called "anti-aliasing." The technique is really quite simple. We decide what size we want our final image to be, and we take our samples at a higher resolution than our final size. So if we want a 100x100 image, we use at least 3 times the number of pixels in our "supersampled" image - 300x300, or 400x400 for even better results.
But wait, we want a 100x100 image, right? Right. So far, we haven't done anything special. The anti-aliasing part comes in when we take our supersampled image and use a filter to combine several adjacent pixels in our supersampled image into a single pixel in our final image. The choice of the filter is very important if you want the best results! Most image manipulation and paint programs have a resize with anti-aliasing option. You can try this and see if you like the results. Unfortunately, most programs don't tell you exactly what filter they are applying when they "anti-alias," so you have to subjectively compare different tools to see which one gives you the best results.
The most obvious filter is a simple averaging of neighbouring pixels in the supersampled image. Being the most obvious choice, it is generally the one most widely implemented in programs. Unfortunately it gives poor results. However, many fractal programs are now beginning to incorporate anti-aliasing directly in the fractal generation process along with a high quality filter. Unless you are a programmer, your best bet is to take your supersampled image and try different programs and filters to see which one gives you the best results.
An example of such filtering in a fractal program can be found on Dennis C. De Mars' web page on anti-aliasing in his FracPPC program: http://members.aol.com/dennisdema/anti-alias/anti-alias.html
The original submission from Rich Thomson is available from http://fractals.iuta.u-bordeaux.fr/sci-faq/aliasing.txt
To read more about Digital Signal Processing, a good but technical book is "Digital Signal Processing", by Alan V. Oppenheim and Ronald W. Schafer, ISBN 0-13-214635-5, Prentice-Hall, 1975.
For more on anti-aliasing filters and their application to computer graphics, you can read "Reconstruction Filters in Computer Graphics", Don P. Mitchell, Arun N. Netravali, Computer Graphics, Volume 22, Number 4, August 1988. (SIGGRAPH 1988 Proceedings).
If you're a programmer type and want to experiment with lots of different filters on images, or if you're looking for an efficient sample implementation of digital filtering, check out Paul Heckbert's zoom program at ftp://ftp.cs.utah.edu/pub/painter/zoom.tar.gz