So, if Alice and Bob are in two spaceships and they fly past each other in different directions at different speeds, each one of them sees the other person’s clock appear to tick slower than theirs.
We have to be a little bit careful with this word “sees”, what they actually “see” is a relativistic Doppler shift, so the clock appears to tick faster as they approach each other and slower as they recede from each other, but if they try to calculate what is happening and form a mental model of the opposite spaceship and correct for the conventional Doppler effect based on the speeds that they observe, I am saying that they conclude that the opposite clock was ticking slower than theirs. (Something similar happens with length contraction, it turns out that anything that casts a circle on your retina, in relativity, will always cast a circle on your retina no matter how fast it is going. So we always illustrate length contraction with these images of balls getting squished as they move faster, but it's important to note that you never see the balls squish with your eyes, you have to try to work backwards to think about what is happening to the ball such that you see what you see, and it is only after you do that that you conclude that in your coordinates the ball must have squished.)
Now you might think that this is a great paradox, two people both thinking that the other person is moving in slow motion. “Why can't Alice just call up Bob, and one of them will be talking slowly and the other will talk faster and they will know who is truly at rest and who is truly moving?”
But now I ask, call them up with what? A phone, you answer. Ah, but that phone works by transmitting light, just a sort of light that is about a foot long and so your eyes are way too small (and not sensitive enough to small energies) to see it. A consequence of using light is that as these two spaceships go closer to the speed of light, and experience more of this time dilation effect, there is also a propagation delay, it takes time for the message from one of them to reach the other. These propagation delay is actually make it impossible to figure out which one of them is correct.
This is more or less why in relativity signals cannot propagate faster than light. Certainly in the instantaneous limit, you would have to choose one of these two reference frames to be “correct” and the other to be “incorrect”, if you had a phone that could connect Alice and Bob without these propagation delays.
Interestingly, such a phone might still lead to Alice hearing Bob talk in slow motion while Bob hears Alice talk in slow motion: but if so, one of those two would have to work backwards and conclude that this phone was sending their messages back in time, which would make them the “wrong” one. So relativity connects things moving faster than light to a sort of time travel. This time travel is not immediately obvious, it requires you to be able to fire faster-than-light bullets in two different reference frames moving very fast relative to each other; you need two “spacelike separations” to sum together into a negative “timelike separation.”
