The Alligator may seem overwhelming at first - it's got quite a few knobs and buttons on its front panel. However, once you've understood the basic signal flow it's actually pretty straightforward. Read through the description below and get familiar with the basics - it will help you a lot when working with the Alligator.
You normally connect the Alligator as an insert effect, so that all of the audio signal passes through the effect device. The incoming signal is split into three, parallel channels. For each channel, there is a separate gate - when that gate is open the signal passes through and when it's closed, the channel is silent. The gates can be opened in four ways:
In this, more detailed diagram, we see that the gate isn't a simple on/off switch - there is actually an amplitude envelope controlling the volume of the channel. When the gate is opened, the envelope is triggered and the sound is let through according to the envelope settings. You can use the amp envelope to soften the attack, to make the notes shorter and more snappy, etc. The gate also triggers a filter envelope, so that each note can get an articulated filter contour. The filter can also be modulated by a global LFO.
Finally, there are Pan and Volume controls. Even a function as basic as stereo panning can make for really interesting, spatial effects - especially since you can pan the three channels, the dry signal and the delay independently!