When I was young, in the back of a car going down the motorway, I was convinced that cars coming the other way sped up as we passed each other. This because when two objects pass each other, at 70mph, from the point of view of one vehicle (it doesn't matter which, due to the symmetry of the situation) the other vehicle seems to be travelling at 140mph.
In an apparently unrelated matter, around the end of the 19th and start of the 20th century, strong evidence was found that the speed of light (in a vacuum) was constant and, importantly, appears the same to all observers.
So if I stand by the side of the motorway and measure the speed of a photon coming towards me, I will find it to be moving at 186,000 miles per second (mps). If I then jump in my car and drive very fast, say 1mps, and then measure the light coming the other way, it doesn't appear to be moving at 186,001mps. Rather, it appears to move at 186,000mps.
Given that speed is equal to distance over time, and the speed is of light is constant, then if we change distance, time must also change to balance the equation. Likewise, if we change our time perspective, then distance must change.
The point lies in the term 'relative.' There becomes no special point of view. The notion of simultaneity breaks down, and two people in different frames of reference (that is, moving at a constant speed relative to one another) can measure the same phenomenon, come up with different measurements, and both be correct.
Note that for these discussions we have to ignore acceleration and gravity, since to include those will require general relativity. Here we will only deal with special relativity.
Imagine a rowing crew running REALLY fast towards a shed. The problem is, the boat they are carrying is longer than the shed. Thankfully, the shed has doors at either end of it. The trick is try and get the whole boat into the shed and have both doors shut at the same time. Because, from the point of view of the shed keeper, the boat is moving really fast, its length is inversely related to its speed and so it appears to contract (known as Lorentz-Fitzgerald contraction); just enough for its new 'apparent' length to be shorter than that of the shed. This allows the shed keeper to shut the first door before he has to open the second door and let them out (remember that they cannot stop, as that would be an acceleration). So both doors are momentarily shut simultaneously.
But the crew do not feel this contraction, and the shed equally seems to be moving towards them at very high speed and hence, contracting. So from the point of the view of the crew, the second door is opened before the first one is shut. This paradox shows the breakdown of simultaneity.