symmetry operation
Suppose someone spins a molecule while your back is turned, then asks whether anything changed. If the molecule looks identical afterward — atom for atom, indistinguishable from before — then the spin they performed was a symmetry operation. It is a move that maps the molecule onto itself: every atom ends up where an identical atom started, so no measurement could ever reveal that the move happened.
Each symmetry operation is carried out using a symmetry element. Rotating by 360/n degrees about a Cn axis is an operation; reflecting through a mirror plane is an operation; inverting every atom through the center i is an operation; doing nothing (the identity E) is, formally, also an operation. The improper operation Sn means rotate, then reflect across a plane perpendicular to that axis, as a single combined move. Operations can be done one after another, and the result of two operations done in sequence is always equivalent to some single operation in the same set — that closure is what makes the collection a mathematical group.
Why bother naming these moves? Because the complete list of operations a molecule allows is its fingerprint. That list (the point group) controls which vibrations show up in infrared versus Raman spectra, whether the molecule can be chiral or carry a dipole, and how its atomic orbitals are allowed to combine into bonds. Far from being abstract geometry, the catalogue of symmetry operations is the practical engine behind much of inorganic bonding theory and spectroscopy.
For water, the operations are: E (do nothing), C2 (rotate 180 degrees about the axis bisecting the H-O-H angle), and two reflections (one in the molecular plane, one perpendicular to it). Carry out any of these and the molecule is indistinguishable from before.
Water's four symmetry operations: E, C2, and two reflections.
The set of all symmetry operations of a molecule forms a mathematical group (it satisfies closure, has an identity E, and every operation has an inverse) — which is exactly why the subject is called "group theory."