Dividers of mirror images always lack a dimension the shape they divide features: A one-dimensional line is always symmetrical with its zero-dimensional center acting as a point of symmetry. Two-dimensional shapes are symmetrical if, when you fold them on their one-dimensional line of symmetry, one half covers the other. In similar fashion, three-dimensional shapes can also be symmetrical and their mirror images must be divided by 2-D shapes.
Suppose you see only one long side of a cuboid prism from the side and draw a line with a marker to divide it into two mirror images. Move your head to watch the prism from above; the line would now be invisible, not allowing you to determine the solid's mirror images. The only way to see where the shape is divided into mirror images from every angle, is to have a thin square inside the prism and this is what a plane of symmetry is.
A plane is a two-dimensional surface, which is flat and has no thickness. As professor David E. Joyce mentions on "Euclid's Elements," a plane can be infinite, which means that its two dimensions can extend forever, but it can also be a finite shape, such as a square or a circle. Theoretically, a plane is where two-dimensional geometry takes place. But you can also use planes to depict two-dimensional shapes in a three-dimensional space, which is why planes are used in reflection symmetry.
Planes of symmetry are two-dimensional shapes that "cut through" 3-D solids, to depict where the solid is divided into two mirror images. Apart from depth, which in planes is theoretically zero, planes of symmetry have equal dimensions with the solids they divide. In other words, the sides of the plane must always touch the faces of the solid. As with 2-D shapes, a solid can be asymmetrical -- not have any planes of symmetry -- or feature infinite planes of symmetry, such as the sphere.