Incoming links: Lorentz covariance
Generally means that he form of the equation does not change if we transform .
This is generally what we want from the laws of physics.
E.g. a Galilean transformation generally changes the exact values of coordinates, but not the form of the laws of physics themselves.
Lorentz covariance is the main context under which the word "covariant" appears, because we really don't want the form of the equations to change under Lorentz transforms, and "covariance" is often used as a synonym of "Lorentz covariance".
TODO some sources distinguish "invariant" from "covariant": invariant vs covariant.
Basically a synonym of Lorentz covariance?
Subtle is the Lord by Abraham Pais (1982) chapter III "Relativity, the special theory" mentions that this fact and its importance (we want the laws of physics to look the same on all inertial frames, AKA Lorentz covariance) was first fully relized by poincaré in 1905.
And at that same time poincaré also immediately started to think about the other fundamental force then known: gravity, and off the bat realized that gravitational waves must exist. general relativities is probably just "the simplest way to make gravity Lorentz covariant".