What does it mean?
Well, first off I am referring to anti-aliasing as it pertains to image rendering on computers, particularly with fonts and video games. If you want more information on aliasing in general,
this link should suffice.

So what is Anti-Aliasing anyway? Its easier to explain with images than in text. Aliasing happens when converting from analog to digital signals. "What?" you say. Don't worry, I am going somewhere. An analog source is of perfect quality. The file does not have a format, it is merely the same as the source. It is analogous to the source. For example, when recording audio, the variances in electrical signal would be the same as variances in air pressure from the sound itself. Digital files can not do this. There is currently no such thing as a vector (math) audio file. (There should be!). It is easier to think of digital files with video. It is not continuous, but rather sequential frames. The same for audio, an audio file is not collections of waves, but rather point samples of the wave. Just as videos are cut up into frames, audio is cut up into frames. Well, not 'frames'. I believe we (techies) call them samples. The more of these samples one takes per second, the higher quality audio file (well, one aspect of audio files). Aliasing is what happens when not enough of these samples are being taken, and the analog wave can not be constructed. It is a type of distortion in audio files.

Look into more:
Nyquist rate,
Digital Audio Files,
Vector Graphics (all Wikipedia links)
When we switch from talking about audio to talking about actual pixels, or images, the problem we see is pixelation. It doesn't necessarily occur from not enough sampling, but rather from the fact that pixels are big and are square. Big? Yeah big. Scientists, computer geeks, and image professionals are making them as small as possible, because of one reason: You cant draw a circle with a square. Or rather, with a bunch of squares. Go draw a circle (or any curved line) in MSPaint. Now zoom in a lot. You see pixels!

This is aliasing distortion in images. But as you can see above, there is a solution. The solution is called (you guessed it) anti-aliasing. It works by tricking our eyes. The human eye does not see as precisely as you think. What our mind does a lot of filtering, editing, and converting of what are two eyes send us to what it thinks it is seeing. Most optical illusions work by messing with out sense of perception of distance. (Creating a false perspective that changes the sizes of things in our mind: Closer thing is bigger thing. But I am not here to talk about optical illusions). Anti-aliasing blurs the curve (jagged edge) by coloring in the image a bit. That bit is generally who much of the curve would cut through the pixel. (if the line would go through 40% of the pixel, it will become 40% darker (40% more the color of the curve)). Generally. Different Anti-aliasing algorithms work in different ways, but that is the technique used. Shade nearby pixels (creating a blur) that our mind puts together as a sharp, clean, curve. Some Anti-Aliasing algorithms, such as Microsoft's ClearType (which is making this text look better right now! Probably!) actually adds color into the b/w mix. The reason that works is more complicated. A tad too complicated for me to get into or bother researching to a level of understanding high enough that I could present it succinctly.
Anti-aliasing is the process of making pixelated and jagged images look smooth and crisp.
Generally, digital designers like anti-aliasing because it makes images look better, and un-jagged, while print designers don't because nobody wants to print a blurry image.
If you stumbled onto this article because you are curious about your video game settings, just remember that the lower the resolution, the higher the need for anti-aliasing. And pixelation/aliasing is not the game developers fault, it's the way computers work.
Here is my research, it's worth reading into more, in my opinion.