To physicists, time is defined by quantum mechanics. A photon with energy h (Planck's constant) behaves as though it were oscillating once per second. That's a definition. Atomic clocks are based on this.
Time direction is something else. It is based on information, which has long sat uneasily in the world of mathematics, because it exists only as a collective property. One quantum by itself can not have a time direction. But, any multi-quantum system must have an arrow of time.
To see this most simply, set up a row of ten numbered balls
The essence of quantization is that information is limited - many 'different' particles are
indistinguishable. We can not distinguish one electron from another, we can only observe the
recent history of each one as reflected in its few quantum numbers. So, repeat the experiment, but
represent balls 1-5 as white, 6-10 as black:
So, with quantization and its consequent limitation of information, a closed universe progresses from an ordered state to a disordered state - a direction of time. And, when the number of particles is large, even as large as the number of molecules in a cubic millimetre of air, the disordered state is permanent, compared to the apparent lifetime of our universe anyway. We can't put things back the way they were, because we can never know how to do it. StarTrek-inspired teletransport of people will face the same limitation.
This, of course, is the 2nd law of thermodynamics - entropy is information with the sign wrong. Space-time is also dissipative with respect to information: the information obtainable from an object moving away from us is progressively reduced as its speed approaches the speed of light; anything heading away from us at the speed of light is no longer observable.
A direction of time is implicit in general relativity. Gravity is asymmetrical and self-reinforcing, and the appearance of a black hole (at this point in time, anyway) is irreversible. So, contrary to the view of many physicists whose formative years were spent solely with classical mechanics, time direction seems to be inherent in all aspects of our universe.
In time, this view may change. (String theorists are trying.) In the meantime, we make do with counting cycles from an arbitrary 'Start of Everything'.
Our earliest measures of time were the duration of one rotation of the earth relative to the sun, a rotation of the moon about a point on the earth, and of the earth around the sun in inertial space, measures based on general relativity. These were built into the genetic heritage of all life long before human beings arose. We still use them today, as our day, month and year. In our Western civil calendar, we keep count of days to mark the passage of time. The month is an artifact of history since it no longer matches moon cycles, and the number of days in a year is fiddled to keep our day time and year time in step. Otherwise, event time was used: when the river floods, when the rains come, when the cows come home, when people feel it's suitable. Much of the world still operates this way.
The earliest clocks, things that measured time by means of physics, were sun dials. They of course only operate in sunny daytime, so events still held sway for the most part. The first true timepieces were water clocks, the time required for a container to empty through an aperture; they were in use in Egypt by 1500 BC. In northern climates where water would freeze, sand was used, or candles. It was not until 1656 that the first pendulum clocks provided time measurement to the minute, and it was not until then that the word 'speed' appeared in the English language. By 1700, pendulum clocks showed that there were variations of about 50 seconds in the duration of a solar day. For the first time, mechanically measured time took precedence over astronomical time; remember, however that both are based upon general relativity.
We now measure seconds by counting cycles derived from quantum mechanics, international atomic clock time (TAI), which is more constant than any mechanical device. The newest clocks, based upon the strontium ions that I pioneered at Canada's National Research Council labs, can be constant to 17 figures. The number of seconds in a day is occasionally fiddled to keep our civil clock time (UTC) in step with the solar day.
We could also measure time by the stars, but we never seem to have done that except for marking seasons of the year. Some birds, however, use them for navigation; perhaps other animals do too.
I am often asked, given my study of time, if time travel is possible. I have 2 comments to offer based on causality:
other notes on physics