Underexposure in photography refers to an image where too little light was recorded. The degree of underexposure will determine how dark a photo is. A slight underexposure can lead to a deepening of the color saturation and this may be a nice effect.

Is it better to over expose or under expose?

If you are shooting JPEG, then the general rule is to underexpose because if you lose the highlights in a JPEG, these highlights are simply lost, unrecoverable. If you are shooting raw, the general rule is to overexpose the image to get more light (more exposure) into the shadows.

What does Shutterspeed do?

Shutter speed is exactly what it sounds like: It's the speed at which the shutter of the camera closes. A fast shutter speed creates a shorter exposure — the amount of light the camera takes in — and a slow shutter speed gives the photographer a longer exposure.