An object is axisymmetric if it has an axis from which the object is the same in any direction, i.e., from any point on the axis, the object is identical in any direction that is perpendicular to the axis. A disk, a sphere or cylinder is axisymmetric but a cube is not.
In astrophysics, a galaxy, a circumstellar disk, ring systems such as Saturn's may be very close to axisymmetric or largely so or generally not. Planets, being basically spherical are quite close. Minor planets such as asteroids vary.