Symmetry means the invariance of the structure of a mathematical or physical object with respect to transformations of the object.
Reflection symmetry. Reflection symmetry, line symmetry, mirror symmetry, mirror-image symmetry, or bilateral symmetry is symmetry with respect to reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry. In 2D there is a line of symmetry, in 3D a plane of symmetry.  generally speaking, a balance or correspondence between various parts of an object; the term symmetry is used both in the arts and in the sciences. In art and design, it is often used in a somewhat loose sense, to mean a kind of balance in which the corresponding parts are not necessarily alike but only similar. (1) A symmetry, in the narrow sense, is a reflection with respect to a plane α in space or with respect to a line a in the plane. Such a symmetry is a transformation of space or the plane that takes every point M into a point M’ such that the line segment MM’ is perpendicular to α or a and is divided in half by α or a. For example, the symmetry of space and time with respect to translation and rotation means that a given experiment should yield the same results regardless of where it is performed, what direction the equipment is pointing in, or when it is performed. (1) A symmetry, in the narrow sense, is a reflection with respect to a plane α in space or with respect to a line a in the plane. Such a symmetry is a transformation of space or the plane that takes every point M into a point M’ such that the line segment MM’ is perpendicular to α or a and is divided in half by α or a. Generally, an object with 'rotational symmetry' also known in biological contexts as 'radial symmetry', is an object that looks the same after a certain amount of rotation. An object may have more than one rotational symmetry; for instance, if reflections or turning it over are not counted. In another meaning of the word, the rotation group of an object is the symmetry group within E + (n), the group of direct isometries; in other words, the intersection of the full symmetry group and the group of direct isometries. For chiral objects it is the same as the full symmetry group. Definition of Line of Symmetry. Line of symmetry is a line that divides a figure into two congruent parts, each of which is the mirror images of the other. When the figure having a line of symmetry is folded along the line of symmetry, the two parts should coincide.  In mathematics, symmetry has a more precise definition, that an object is invariant to a transformation, such as reflection but including other transforms too. Although these two meanings of symmetry can sometimes be told apart, they are related, so they are here discussed together. This means that an object is symmetric if there is a transformation that moves individual pieces of the object but doesn't change the overall shape. The type of symmetry is determined by the way the pieces are organized, or by the type of transformation: Symmetry comes from a Greek word meaning 'to measure together' and is widely used in the study of geometry. Mathematically, symmetry means that one shape becomes exactly like another when you move it in some way: turn, flip or slide. For two objects to be symmetrical, they must be the same size and shape, with one object having a different orientation from the first. There can also be symmetry in one object, such as a face. If you draw a line of symmetry down the center of your face, you can see that the left side is a mirror image of the right side. Not all objects have symmetry; if an object is not symmetrical, it is called asymmetric. (3) Symmetry, in the general sense, means the invariance of the structure of a mathematical or physical object with respect to transformations of the object. For example, the symmetry of the laws of the theory of relativity is determined by their invariance with respect to the Lorentz transformations. (1) A symmetry, in the narrow sense, is a reflection with respect to a plane α in space or with respect to a line a in the plane. Such a symmetry is a transformation of space or the plane that takes every point M into a point M’ such that the line segment MM’ is perpendicular to α or a and is divided in half by α or a. Sometimes called line symmetry or mirror symmetry, reflection symmetry is when an object is reflected across a line, like looking in a mirror. The face from before shows you reflection symmetry. Here are some more examples of reflection symmetry. There can also be symmetry in one object, such as a face. If you draw a line of symmetry down the center of your face, you can see that the left side is a mirror image of the right side. Not all objects have symmetry; if an object is not symmetrical, it is called asymmetric. more ... Symmetry is when one shape becomes exactly like another if you flip, slide or turn it. The simplest type of Symmetry is Reflection (or Mirror) Symmetry, as shown in this picture of my dog Flame. There is also Rotational Symmetry and Point Symmetry. Symmetry-Reflection and Rotation. 
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what symmetry means