Lorentz covariance is a fundamental principle in the theory of relativity that describes how the laws of physics remain invariant under Lorentz transformations, which relate the coordinates of events as observed in different inertial reference frames moving at constant velocities relative to each other. In more detail, Lorentz transformations include combinations of rotations and boosts (changes in velocity) that preserve the spacetime interval between events.
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Lorentz covariance means that a physical equation retains its form when coordinates and the components of its fields are transformed consistently by a Lorentz transformation. For a four-vector, the component rule is ; derivatives and other fields must transform by their own corresponding rules.
Covariance is not the assertion that every component has the same numerical value in every frame. The four-momentum changes components while its contracted Minkowski norm is invariant. Nor does covariance of a differential equation uniquely choose a solution: initial conditions, boundary conditions and any independent homogeneous solution must also be specified consistently. This distinction is important when translating fields and observables between inertial frames.
Same motivation as Galilean invariance, but relativistic version of that: we want the laws of physics to have the same form on all inertial frames, so we really want to write them in a way that is Lorentz covariant.
This is just the relativistic version of that which takes the Lorentz transformation into account instead of just the old Galilean transformation.