Caption: An illustration of Time dilation (more precisely, special relativistic time dilation) and the asymmetry in time dilation between the situation with the two clocks in Frame A (which is an inertial frame) and the one clock in Frame B (which is another inertial frame). Frame A and Frame B are NOT accelerated with respect to each other.
Features:
By "time flows differently", we mean as measured by all clocks---mechanical clocks, atomic clocks, biological clocks---all clocks.
The effect is called time dilation and it is probably the most mind-blowing feature of special relativity.
A Frame A observer and a Frame B observer both agree on this.
But they also agree to observe the effect that there must be two clocks in Frame A at different places in Frame A and one clock in Frame B.
This is the asymmetry between the two observers that allows an asymmetric effect. Otherwise one might wonder why both observers see the other's clock as running slow which is symmetrical but impossible.
Without mathematical detail, it is hard to explicate how the asymmetry in number of clocks leads to asymmetry in time measurements. However, one can say that it an aspect of the connection between space and time in special relativity. The relativity speak term spacetime emphasizes the connection.
In this case, there is another kind of asymmetry, since special relativity does UNDER RECONSTRUTION BELOW
Time is literally running slower in those inertial frames.
The effect grows as the relative speed grows.
Now constant velocity motion is relative in special relativity, and so two observers in relatively moving inertial frames would both measure clocks in the other observer's frame as running slow.
This is paradoxical.
But, in fact, the paradox is resolved by taking space as well as time effects into account.
The full explication is NOT so a hard, but a bit too intricate.
But the partial explication is easy.
Observer 1 measures their own clock at ONE place in their reference frame, but measures observer 2's clock at TWO or more places in their reference frame (i.e., observer 1's reference).
So there is an asymmetry between the measurements for observer 1.
Observer 2 has the same asymmetry for their measurements.
Since the asymmetry is the same, the observers both observe the same effect: the moving clock runs slow relative to the observer observing the clock in motion.
But what about the twin paradox? See the explication below in subsection The Twins Paradox.