The principle of relativity is not in itself new or something strange and unfamiliar, but goes back to the physics of Galileo and Newton. It expresses the common experience that some aspects of the world look different to observers who are in motion relative to each other. Thus, somebody on the ground following a bomb released from an aircraft will watch it describe a steepening curve (in fact, part of an ellipse) in response to gravity, while the bomb aimer in the plane (ignoring air resistance) sees it as accelerating on a straight line vertically downward. Similarly, they will perceive different forms for the path followed by a shell fired upward at the plane and measure different values for the shell's velocity at a given point along it.
So who's correct? It doesn't take much to see that they both are when speaking in terms of their own particular viewpoint. Just as the inhabitants of Seattle and Los Angeles are both correct in stating that San Francisco lies to the south and north respectively, the observers on the ground and in the plane arrive at different but equally valid conclusions relative to their own frame of reference. A frame of reference is simply a system of x, y, and z coordinates and a clock for measuring where and when an event happens. In the above case, the first frame rests with the ground; the other moves with the plane. Given the mathematical equation that describes the bomb's motion in one frame, it's a straightforward process to express it in the form it would take in the other frame. Procedures for transforming events from the coordinates of one reference frame to the coordinates of another are called, logically enough, coordinate transforms.
On the other hand, there are some quantities about which the two observers will agree. They will both infer the same size and weight for the bomb, for example, and the times at which it was released and impacted. Quantities that remain unvarying when a transform is applied are said to be "invariant" with respect to the transform in question.
Actually, in saying that the bomb aimer in the above example would see the bomb falling in a straight line, I sneaked in an assumption (apart from ignoring air resistance) that needs to be made explicit. I assumed the plane to be moving in a straight line and at constant speed with respect to the ground. If the plane were pulling out of a dive or turning to evade ground fire, the part-ellipse that the ground observer sees would transform into something very different when measured within the reference frame gyrating with the aircraft, and the bomb aimer would have to come up with something more elaborate than a simple accelerating force due to gravity to account for it.
But provided the condition is satisfied in which the plane moves smoothly along a straight line when referred to the ground, the two observers will agree on another thing too. Although their interpretations of the precise motion of the bomb differ, they will still conclude that it results from a constant force acting in a fixed direction on a given mass. Hence, the laws governing the motions of bodies will still be the same. In fact they will be Newton's familiar Laws of Motion. This is another way of saying that the equations that express the laws remain in the same form, even though the terms contained in them (specific coordinate readings and times) are not themselves invariant. Equations preserved in this way are said to be covariant with respect to the transformation in question. Thus, Newton's Laws of Motion are covariant with respect to transforms between two reference frames moving relative to one another uniformly in a straight line. And since any airplane's frame is as good as another's, we can generalize this to all frames moving uniformly in straight lines relative to each other. There's nothing special about the frame that's attached to the ground. We're accustomed to thinking of the ground frame as having zero velocity, but that's just a convention. The bomb aimer would be equally justified in considering his own frame at rest and the ground moving in the opposite direction.
Out of all the orbiting, spinning, oscillating, tumbling frames we can conceive as moving with the various objects, real and imaginable, that fill the universe, what we've done is identify a particular set of frames within which all observers will deduce the same laws of motion, expressed in their simplest form. (Even so, it took two thousand years after Aristotle to figure them out.) The reason this is so follows from one crucial factor that all of the observers will agree on: Bodies not acted upon by a force of any kind will continue to exist in a state of rest or uniform motion in a straight lineeven though what constitutes "rest," and which particular straight line we're talking about, may differ from one observer to another. In fact, this is a statement of Newton's first law, known as the law of inertia. Frames in which it holds true are called, accordingly, "inertial frames," or "Galilean frames." What distinguishes them is that there is no relative acceleration or rotation between them. To an observer situated in one of them, very distant objects such as the stars appear to be at rest (unlike from the rotating Earth, for example). The procedures for converting equations of motion from one inertial frame to another are known as Galilean transforms. Newton's laws of motion are covariant with respect to Galilean transforms.
And, indeed, far more than just the laws of motion. For as the science of the eighteenth and nineteenth centuries progressed, the mechanics of point masses was extended to describe gravitation, electrostatics, the behavior of rigid bodies, then of continuous deformable media, and so to fluids and things like kinetic theories of heat. Laws derived from mechanics, such as the conservation of energy, momentum, and angular momentum, were found to be covariant with respect to Galilean transforms and afforded the mechanistic foundations of classical science. Since the laws formulated in any Galilean frame came out the same, it followed that no mechanical experiment could differentiate one frame from another or single out one of them as "preferred" by being at rest in absolute space. This expresses the principle of "Galilean-Newtonian Relativity." With the classical laws of mechanics, the Galilean transformations, and the principle of Newtonian relativity mutually consistent, the whole of science seemed at last to have been integrated into a common understanding that was intellectually satisfying and complete.