The shear center of an arbitrary shape is the point through which a shear (vertical; parallel to the plane of the shape) force can be applied without inducing a rotation.
If that arbitrary shape is a cross-section of a three-dimesional beam, the shear center can be extruded into a line extending through the length of the beam - this line is the elastic axis. You can find one of the elastic axes of your pencil by balancing it on your finger.
For a rectangular section, the shear center coincides with the geometric centroid.
The shear center can also be defined in terms of shear flow: it is the point about which the sum of the moments due to the shear flow induced by a force through itself is equal to zero. That says the same thing as the first paragraph, only in a more complicated manner.
The shear center is not necessarily the same as the center of gravity, though they do have some things in common.
Shear center is widely used in statics, dynamics, and structural analysis, because any force acting through a section can be decomposed into the sum of a force through the shear center and a moment around the shear center. This is sometimes known as the method of superposition, and makes solving such problems much easier.