a symmetry operation

Pick up a square tile and turn it a quarter turn. If you had closed your eyes, you could not tell it had moved: corners land on corners, edges on edges. That 'do something, yet nothing looks different' move is a symmetry operation. It is the basic verb of crystallography, an action you perform on an object that leaves it indistinguishable from how it started.

A symmetry operation is a rigid motion, a rotation, a reflection in a mirror, an inversion through a point, or just leaving things alone (the identity), that carries every atom onto an atom of the same kind so the pattern maps exactly onto itself. Combine two operations and you get a third; every operation can be undone; there is always the do-nothing identity. Those closure rules are exactly the rules of a mathematical group, which is why symmetry is described by group theory. For numbers: a 4-fold rotation about a square's centre is a 90 degree turn, and doing it four times (90, 180, 270, 360 degrees) returns you to the start.

Point operations (rotation, reflection, inversion, rotoinversion) leave at least one point fixed and build the 32 crystallographic point groups. Add the operations that also slide the pattern (translations, screw axes, glide planes) and you get the 230 space groups. Watch the distinction: the operation is the action; the geometric thing it acts about (the axis, the plane, the point) is the symmetry element.