Symmetry is a slippery word to define. If you look it up in most dictionaries, you'll find references to mirrors (symmetry of reflection) or crystals (symmetries like mirror symmetry, but in multiple directions).

But symmetry in physics is used in a much broader sense.

Hermann Weyl : "a thing is symmetrical if one can subject it to a certain operation and it appears exactly the same after the operation."

We should hope that any physical rules we discover obey

translational symmetry : they should behave the same anywhere in space.

rotational symmetry: the orientation of an axis system should make no difference.

Richard Feyman notes some possible "Symmetry Operations"

Translation in space

Translation in time

Roatation through a fixed angle

Uniform velocity in a straight line (Lorentz transformation)

Reversal of time

Reflection of space

Interchange of identical atoms or identical particles

Quantum-mechanical phase

Matter-antimatter (charge conjugation)

Furthermore, Feyman notes, each of these symmetries, when understood at a deep level, implies a Conservation Law. These laws are connerstones for our understanding of the world.