Mathematically, the physical quantities must transform covariantly, that is, under a certain representation of the group of coordinate transformations between admissible frames of reference of the physical theory.
The principle of covariance does not require invariance of the physical laws under the group of admissible transformations although in most cases the equations are actually invariant.
However, in the theory of weak interactions, the equations are not invariant under reflections (but are, of course, still covariant).
In Newtonian mechanics the admissible frames of reference are inertial frames with relative velocities much smaller than the speed of light.
Time is then absolute and the transformations between admissible frames of references are Galilean transformations which (together with rotations, translations, and reflections) form the Galilean group.
The covariant physical quantities are Euclidean scalars, vectors, and tensors.
The transformations between frames are the Lorentz transformations which (together with the rotations, translations, and reflections) form the Poincaré group.
The covariant quantities are four-scalars, four-vectors etc., of the Minkowski space (and also more complicated objects like bispinors and others).
An example of a covariant equation is the Lorentz force equation of motion of a charged particle in an electromagnetic field (a generalization of Newton's second law) where