To begin this discussion, you must understand what an Inertial Reference Frame is. It is a place anywhere wherein anything experiences no other force being applied to it as it moves. But why is this important to the understanding of relativity?

The Inertial Observer is basically anyone who can see something happening, possibly making measurments on what is happening, and is in a state of motion even when he or she is at rest. If there are more than one observer, and they get the same results from measuring the material or force in question, then the amount being measured is considered invariant. Invariance means that although the observers may be seeing different things in respect to what they're measuring (as in the motion or appearance may be differnt), the essential quantities (like speed, weight, etc.) of whatever is being measured will not change.

But let us go to an example:

For a train moving at an unchanging speed:
On the train, a passenger drops a ball in front of himself on the floor. His companion, who sees the ball fall, would see exactly what he would see if the ball were dropped outside, standing unmoving on the ground. He sees the ball fall in the straight, vertical line. The ball drops on the floor in front of the first passenger.

Now, instead imagine yourself as an observer outside the train, at rest with respect to the Earth (moving at the same speed), and you can see into the train. You watch the passenger drop the ball as the train thunders past you. What is different compared to the first example?

Outside the train, you measure the path of the falling ball, in respect to your own stationary postion, to be a parabola.
(Note: The animations do not show this observance.)

The motion of the ball would appear to be a parabola because the ball has the same horizontal speed as the train. Both you and the seated person on the train would agree that the ball fell to the floor because of gravity, and both of you would agree on how fast gravity caused the ball to fall (9.8 m/s²), the mass of the ball, and how long it took for the ball to fall.

But you would disagree on the velocity -- the speed of the falling ball, and the path it took (what it looked like as it fell), as well as the ball's final resting place. These differences occur because of the train's motion. Gravity, mass, and time are quantities that don't change. But the disagreement is because you see the same thing from different places and under different circumstances. You also get differences for the fallen position of the ball and its total speed. This is because the place you observe from, the resting place of the ball, and its total speed are relative quantities, meaning that they are different for an observer in one place compared to the observer in another. So you would see these quantities as different because of your position relative to the observer on the train.

This example assumes that space and time are absolute -- space and time intervals of say 1 second of time and 1 meter of distance are equal in all reference frames (all frames or places are equal because there no frame is better than any other).

Galilean Transformations are the equations that relate the differences in measurements between frames -- they adjust the velocity recorded by the two observers in the ball and train example by the relative velocity between the two. With this example, if the passenger instead threw the ball horizontally, you would measure the ball's velocity as the combination of the train's velocity plus the velocity of the ball measured by the observer on the train.

This shows how transformations work: the ball's measured velocity must be added to the train's velocity for you to measure the velocity in your frame outside on the ground. These measurements would apply anywhere or anytime.
Newtonian Space and Time explains more about this.

Newton's Laws of Motion and Gravity

• Anything anywhere (especially in space) would move in a straight line unless something else changes its direction.
• The new acceleration of this something when it is acted upon is directly proportional to the amount of force acting on it. (Acceleration = change in speed or direction with time)
• Anywhere in the Universe, no matter how far apart, two objects will experience a gravitational attraction that is proportional to the product of the object's masses and inversely proportional to the square of how far apart they are.

There is one oddity that you should remember about reference frames though. For although Newton's laws should apply universally, meaning everywhere, for there is no experiment that can tell one reference frame from another. You can never be able to tell if you are really actually moving, or really at rest, so long as the motion you DO have is unaccelerated (not speeding up). This means you're not moving faster than anything else, otherwise it would be easy to tell frames apart.