To answer a bit about PWM from an earlier post, it stands for Pulse Width Modulation. It's a pretty simple concept: it's where a signal (the pulse) has a set period of time (the width) for which it is active before being shut off (modulation). This event occurs rapidly to create multiple 'pulses.'
In how it would help with fading an LED, each 'pulse' is a signal equal to a certain voltage (in the case of an Arduino, 5 volts). If we are at half the maximum PWM, meaning per instance of a pulse, half the time the pulse is on, and half the time it is not, the LED would light up half as much as it would normally. If we set it to the maximum, then the pulse signal is 'solid' and the LED will illuminate all the way. So we we use the Arduino to cycle back and forth from the lowest to the highest point for the PWM signal, we can make the LED fade.
That's just an extreme nutshell explanation. 90% of the forum could explain it a lot better and in greater detail, but that should at least provide a basic explanation.
-Flame