I am new to camcorders and unfamilar with this vocabulary. What are these two options used for exactly?

In photography, shutter speed refers to how long the shutter is left open. If the shutter is 1/60th of a second, then the shutter allows light to hit the light sensing area (the film) for that long. In video it's similar although I don't think there are any moving parts.

Shutter speed affects exposure and the amount of motion blur the camera picks up. High (fast) shutter speeds are useful for certain special FX (since high shutter speeds remove nearly all motion blur).

Gain is electronical amplification of the signals the CCD puts out. It makes the image brighter but also amplifies and adds electrical noise. In low light situations you may need to use the camera's gain.

There's 2 types of gain. One was already explained by Glenn, the other one is audio gain.

Gain is merely simple signal amplification. Unfortunately, noise is also amplified at the same time.