One of the paradigm shifts in Einstein's revolutionary relativity was the integration of time into our understanding of the structure of space. It was a theory of spacetime. While time had a unique position in that theory, string theory seems to have sundered it once again from the spatial dimensions. The number of spatial dimensions is constrained by physical consequence. Time, as always, is unmotivated and without explanation.
What is time?
Sometime early in our childhood, we become aware that events do not happen simultaneously. There is a sequence to events. That separation between events is time.
Einstein enshrined this principle, noting that the constancy of the speed of light provided us a convenient yardstick for time. If two events are related as cause and effect, the shortest possible interval between them is defined by the time that a light-beam would require to travel from one to another.
But then, why does time seem to run in one direction only? Einstein's principle does not provide a directionality to time: it would seem that, if we reversed the direction of the interacting particles, the reversed sequence would be as amenable to physical analysis as was the original sequence. In fact, the mathematical equations of classical physics and relativity do not prefer a direction.
Classically, asymmetry in the flow of time is explained by the non-uniformity of the distribution of matter and energy. The microscopic consequences of any event cause a redistribution of energy that tends to disorder. That disorder spreads into the environment, and some of the energy will escape into the vacuum of outer space, never to be seen again. Under current theory (my earlier, wild speculations aside) only at the end of the universe, when all matter is cooled to the temperature of the cosmic microwave radiation, will time-reversal of events be at all probable.