The bridge is where the strings cross before they are attached to the body. The bridge holds the stings where they are supposed to be and keeps them there. The bridge on the left is a "Floyd Rose Tremolo" that can be used to make different sounds from bending notes. The bridge on the right is a fixed bridge and has no movement to it at all. A person would pick different bridges when considering the cost and sound of a guitar.

Both pictures above courtesy of Carvin Guitar Company