Galilean invariance is a principle which states that the fundamental laws of physics are the same in all inertial (uniform-velocity) frames of reference. Galileo Galilei described this principle using the example of a ship traveling at constant speed, without rocking, on a smooth sea: someone doing experiments belowdecks would not be able to tell whether the ship was moving or stationary.
Specifically, the term Galilean invariance today usually refers to this principle as applied to Newtonian mechanics, under which all lengths and times remain unaffected by a change of velocity, which is described mathematically by a Galilean transformation.
Maxwell's equations governing electromagnetism possess a different symmetry, Lorentz invariance, under which lengths and times are affected by a change in velocity, which is then described mathematically by a Lorentz transformation. Albert Einstein's central insight in formulating special relativity was that, for full consistency with electromagnetism, mechanics must also be revised such that Lorentz invariance replaces Galilean invariance. At the low relative velocities characteristic of everyday life, Lorentz invariance and Galilean invariance are nearly the same, but for relative velocities close to that of light they are very different.
