Galileo's principle of relativity states that the laws of mechanics are invariant in every inertial frame of reference.
This is well illustrated by Galileo’s ship. What is meant here by "laws of mechanics"? Are these Newton's laws of motion, conservation laws, Lagrange/Hamilton equations, or something else?
Galileo's principle of relativity states: "It is impossible by mechanical means to say whether we are moving or staying at rest". This predates but underpins Newton's laws of motion. It states that the basic principles governing the motion of objects (which would later be formalized in Newton's laws) apply equally in all inertial frames (frames that are either at rest or move with a constant velocity).
Galileo did not specifically mention Newton's laws, conservation laws, or the formulations of mechanics by Lagrange and Hamilton as he preceded all of these developments, however, his principle implies that these fundamental laws of mechanics are invariant across different inertial frames.
This principle is significant because it introduces the idea that the laws governing physical phenomena are consistent and universal, regardless of the observer's state of motion, laying the groundwork for classical mechanics and influencing future developments in physics, including Einstein's theories of relativity.
By "laws of Physics," we should mean the equations describing physical phenomena. Their invariance does not mean that the same phenomenon is described in the same way in different reference frames. This is clearly wrong. It means that if we prepare our experiments in the same way in different reference frames the invariance of the equations implies the same solutions and the practical impossibility of deciding only on the basis of the phenomena what the motion of the reference frame is.
From the historical side, one should distinguish between what we call Galilean invariance and what Galileo meant by his ship argument.
Galilean invariance is limited to the classical mechanics formalism. Galileo's ship argument has a much wider meaning. Even if mechanics dominated the physics of Galileo's time, the ship argument was not limited to mechanical phenomena. When Galileo wrote (translation as in this Wikipedia page )
... The fish in their water will swim toward the front of their bowl with no more effort than toward the back, and will go with equal ease to bait placed anywhere around the edges of the bowl. Finally the butterflies and flies will continue their flights indifferently toward every side, nor will it ever happen that they are concentrated toward the stern, as if tired out from keeping up with the course of the ship, from which they will have been separated during long intervals by keeping themselves in the air. And if smoke is made by burning some incense, it will be seen going up in the form of a little cloud, remaining still and moving no more toward one side than the other. The cause of all these correspondences of effects is the fact that the ship's motion is common to all the things contained in it, and to the air also.
He claimed that no phenomenon allows the distinction of the motion of two reference frames in uniform relative motion. Flies, butterflies, and fish are not exclusively mechanical systems. Still, they behave the same way.
Galilean invariance means that in going between different frames of reference, velocities are additive. For example, the velocity of a baseball thrown from a moving car is the sum of the ball's velocity relative to the car and the car's velocity relative to the ground. This formalism works for velocities much smaller than the speed of light.
It fails for very high velocities and is replaced by relativistic invariance, in which velocities are not purely additive. In this case, the velocity of a beam of light emitted by the headlight of a moving car is not added to that of the car; it is always c, no matter how fast the car is moving.
It might be of your interest to look at this recent paper of mine precisely on the question of what, exactly, Galileo's principle of relativity says.
Very briefly, this paper makes the case that there really are two different principles that go by the name "Galilean principle of Relativity", and physicists have tended to conflate them. One is the principle mentioned by @cconsta1 and @giorgio that requires repeating the experiments with the same initial conditions in different initial frames. If you do that, this principle says that you will get the same outcomes regardless of the (constant) velocity of the frame (the paper calls it "internal Galilean relativity principle").
The other principle looks into the mathematical invariance of the laws of a system under Galilean transformations (and Galilean boosts in particular). When physicists say "the laws of classical systems must be Galilean invariant", this is often the principle they have in mind. But this second principle was not really formulated by Galileo (who preceded Newtonian mechanics), and it is logically independent from the first one. (you cannot logically derive one from the other one).
For example: Hooke's law (or more precisely, the law of the classic harmonic oscillator, $x=-Cx''$) for an ideal spring is NOT Galilean invariant, but of course you will get the same outcomes playing with a spring regardless of the constant velocity of the frame where the spring is situated (provided the initial conditions are the same). In fact, the laws of many systems (classical or not) are not strong text Galilean invariant, and yet they still satisfy the first version of the relativity principle explained here. Think of the law for a classical string attached to two walls, which is not invariant under Galilean boosts as normally written, or think of sound waves as well, which acquire additional terms when the frame is boosted. The paper also has a historical overview of these principles if you are interested in that. Pay special attention to the fact that many physicists, including Einstein, formulated the principle of relativity different in different texts.
EDIT: I see this response has generated a lot of discussion. I want to repeat my main point, namely, that physicists use the term "Galilean Relativity" or the term "Principle of Galilean Relativity" to refer to two different principles, not always properly distinguished from one another. The paper I mention above shows a lot of textual evidence that this is, in fact, the case (just open some textbooks and you will see the differences). In fact, one does not need any fancy math to see the overall point. Even if we do not know what differential equations model the behavior of a fish in a bowl inside a ship, we know, as Galileo pointed out, that the fish will behave in the same manner inside the bowl regardless of the constant velocity of the ship with respect to the shore. To know this we do not need to check if the equation for the fist (whatever it is!) is invariant under Galilean boosts or not (in fact, it probably is not invariant). Hence, we must distinguish the invariance of the equation (Galilean invariance) from satisfaction of the principle of Galilean Relativity.
