In this summary from the eBook Relativity 4 Engineers, the question "what is relativity?" is answered in a non-technical way. The eBook treats the same topic in depth.
The Principles of Special Relativity
Einstein was lead to his special theory of relativity by his believe that there is no way to detect absolute motion. This dictated that the measured speed of light must be the same in all inertial frames of reference. If this was not so, the laws of physics would have been different in inertial frames that move relative to each other.
Inertial frames are uniformly moving coordinate systems, far away from gravitational effects or any other form of influence, where inertia is isotropic, meaning a given force will cause the same acceleration on identical masses in whatever direction the force is applied.
The Aether Abolished
Einstein also argued that if there was a 'rest-frame' for light (the luminiferous aether), we could in principle set up an inertial frame in which light would not propagate in the forward direction at all, e.g., if the frame moves at the speed of light relative to the aether. Einstein was still very young, when he reportedly contemplated if he would still be able to see his own face in a mirror if they were both at rest in such a frame - moving at the speed of light through the aether.
Einstein realized that it is paradoxical to assume the same light ray can actually move with the same speed c (in an absolute Newtonian sense) relative to all inertial frames. This would require that light adapt its "absolute speed" to the frame that measures it. He decided that either time intervals or distance intervals (or both) must change if measured by observers in different inertial frames in relative motion.
The Spacetime Interval
In Newtonian dynamics, space intervals and time intervals are the same for all inertial observers, no matter how fast they move. In Einstein's dynamics, it is the spacetime interval that remains the same for all inertial observers, no matter how fast they move. This concept offers the best answer to the question: "what is relativity?"
The square of the spacetime interval is the difference between the square of the time interval and the square of the distance interval, given that all are expressed in the same units. Time is converted to a distance by multiplying it with the speed of light, e.g., one second represents 300 million meters (the speed of light is 300 million meters per second).
Let us calculate a simple example. Suppose you ride on-board a spaceship that travel from Earth to Alpha Centauri at a speed of 80% of the speed of light. The trip takes 3 years on your calendar. But wait, Alpha Centauri is about 4 light-years from Earth, so how could you possible have reached it in 3 years? To answer the question, consider your spacetime interval between the two events - your departure from Earth and your arrival at Alpha Centauri.
The time interval on your calendar is 3 years. The space interval in your reference frame is zero. How can this be? Well, you were present at both events (departure and arrival), so in your frame of reference, there cannot be a space interval between the two events! So what is the spacetime interval? It must be 3 light-years, because you subtract zero-squared from 3-squared, leaving you with 3-squared.
Now this spacetime interval must remain constant for all observers, thus also for Earth-bound observers. From Earth, we measured the distance to Alpha Centauri accurately as 4 light-years, which must then be the space interval between your departure from here and your arrival at Alpha Centauri. We know that the spacetime interval is 3 years for everyone, so what time interval does it give us? We take 3-squared and add 4-squared, giving 25. Take the square root and we have the time interval for Earth's inertial frame: 5 years.
So, after all, you did not travel at faster than light! In our reference frame, you traveled at our original assumption of 80% of the speed of light (4 light-years in 5 years). If we, erroneously, take our distance and divide it by your time, we would end up with 4 divided by 3, or a speed of 133% of light-speed. If you insist that you want to know the distance between Earth and Alpha Centauri in your inertial reference frame, you can take your measured speed (0.8c) and multiply it by your measured time interval (3 years) and get a distance of 2.4 light-years.
Time Dilation and Length Contraction
In trying to answer the question "what is relativity?", we have indirectly 'stumbled upon' the phenomena of time dilation and length contraction due to velocity. The constancy (or invariance) of the spacetime interval is underlying to relativistic time dilation and length contraction. With your relative speed of 0.8c, your time dilation factor is 0.6, meaning you will measure times and radial distances that are only 60% of what we on Earth would have measured.
The reason for belaboring the observation of the time- and space-intervals between events is this: events in empty space give us something 'tangible' to base special relativity comparisons between inertial frames upon. It does not say whose clock is running faster or slower than anybody elses, but it does say unequivocally who will measure the shorter time and distance between two events---it the observer who is present at both events.
This answers a big part of the topical question: what is relativity?
You will find more depth in the answer to "what is relativity?" in the first part of chapter 1 of the eBook Relativity 4 Engineers, linked to below.
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