A symmetry is (in most classical situations) invariance under a group action, or infinitesimally, invariance under a Lie algebra action. Indeed, historically the mathematical term “group” is a contraction of group of symmetries (e.g. Klein 1872).
In physics, see local symmetry, global symmetry, spontaneous symmetry breaking.
Felix Klein, Vergleichende Betrachtungen über neuere geometrische Forschungen (1872)
translation by M. W. Haskell, A comparative review of recent researches in geometry , trans. M. W. Haskell, Bull. New York Math. Soc. 2, (1892-1893), 215-249. (retyped pdf, retyped pdf, scan of original)