Special relativity is an incredibly simple model of physics that leads to very strange (to us) conclusions.
Relativity was nothing new when Einstein published his first paper on Special Relativity. Centuries before, Galileo had realised that motion made sense only as being relative to something else. For example, a ball could be thrown and caught easily within a ship even if that ship was sailing at speed.
What Einstein did was to add to relative motion the idea of a limiting speed, which would be the same for everyone no matter how fast they were moving. The idea of this limiting speed came from the work in the 1800's on electromagnetism, as there was a constant relating electric charges to magnetic forces, and that constant involved the speed of electromagnetic waves - the speed of light.
Einstein insisted that electromagnetism (and everything else about physics) should work the same no matter how fast you are moving relative to anything else - you should not be able to perform an experiment to determine some absolute speed at which you were moving.
So, if the speed of light was the same for everyone, what would be the result? It would be that speeds would not add up simply. If someone travelling at 20km/h turned on a torch and projected the beam of light ahead, both that person, and someone stationary relative to that person, should measure the speed of the beam as the speed of light (not the speed of light + 20km/h). A vast number of experiments show that this is the case. Other effects would be that lengths of objects moving close to the speed of light will appear shortened, and time should appear to move more slowly for such objects. These effects have all been measured.
What with everything changing depending on speed, is there something that everyone will agree on? Yes, there is - it is called the 'spacetime interval' between two events. You may remember that the distance d between two points A and B in space is given by sqrt((Ax - Bx)2 + (Ay - By)2 + (Az - Bz)2): the square root of the squares of the distances along three axes between A and B.
Special Relativity deals with points in 'spacetime' - the 4-dimensional combination of space and time (known as 'Minkowski space'). However, Relativity also has to deal with time being a different kind of dimension than those of space. This difference results in changes of signs. The equivalent of distance in Minkowski space is the 'spacetime interval' s, and it's calculated like this
s = sqrt((Ax - Bx)2 + (Ay - By)2 + (Az - Bz)2 - c2(At- Bt)2 )
It's the normal Pythagorean calculation, but with the difference in time squared multiplied by the square of the speed of light subtracted.
A speed (the speed of light) comes into the equation because it converts the units of time to units of distance.
No matter how something is moving, all observers will get the same value of 's' for that thing. That's what stays the same in Einstein's Special Relativity.
By the way, you might be wondering what happens if the term involving times gets bigger that the sum of the terms involving space, giving a negative number to be square rooted. This gives a perfectly acceptable 'imaginary number'. 'imaginary' spacetime intervals connect events which can be causally connected - one event could possibly be the cause of another. Non-imaginary intervals are between events which cannot be causally linked, as faster-than-light speeds would be needed to get from one event to the other.