What is a remote shutter release? In what kind of photo situation would I use a shutter release?
Anything you can use to trigger the camera shutter without touching it. :o)
Serious. It can be a remote or cable based control for your camera shutter. It's main advantage is allowing you to take shots without interfering with the camera stability, but it could also be used for shooting from awkward/distant positions or when taking shots including yourself.
Another common use of them is to do aerial photography (using R/C planes and helicopters, kites etc), where the shutter can be controlled by radio or electrical signals through a wire.
A third option, but not exactly remote, is to use automatic shutter control mechanisms based on time (those are usually available in the camera itself) or events. Using special software (for example CHDK for Canon cameras) or tethering (with a computer or using Triggertrap for example) you could make the shutter trigger whenever there is movement in the scene or with the external trigger, a change of light, time or distance intervals or some other event.
It's a (wired or wireless) button to release the shutter. There are a couple of times they're useful. One is for things like shooting wildlife in a situation where you can set the camera up close to a likely spot, and trigger it remotely when a good target walks/flies/swims/whatever into its field of view.
Another is when you want to minimize camera shake for longer exposures, when pushing the button on the camera itself would tend to shake the camera a bit, but little or no vibration is transmitted to the camera by a remote release.