We know the definition of "time" of Augustine of Hippo:
"If no one asks me, I know: if I wish to explain it to one that asketh, I know not" (Conf.).
What is time? Philosophically, what can be said about time in modern times? Moreover, does time exist independently of men, being that man is the only one who can talk about it?
Time is as much a mystery for us as it was for St Augustine. But science and philosophy have sharpened the questions. Thanks to statistical mechanics, we can now formulate the problem of the direction of time. Thanks to general relativity, the science of spacetime, we can now rigorously investigate questions about time travel, branching time, and so on. And thanks to philosophy, we understand the logical geography better: for instance, we know that time might be absolute, relational, conventional, tensed or tenseless, or unreal.
So, although time is still a great mystery, we have definitely progressed from Augustine. We know better what we don't know.
This comes from the last page of Introducing Time: A Graphic Guide by Craig Callender (text), who also edited the, rather more advanced, The Oxford Handbook of Philosophy of Time (blurb: "The first comprehensive book on the philosophy of time"). It fits in your back pocket and has nice pictures too (by Ralph Edney). I enjoyed it very much.
I advocate Mulla Sadra's definition of time. A Persian religious scholar of 17th century who was also a genius of the Islamic tradition of Peripatetic, Neo-platonic Philosophy. He introduced innovative theories of epistemology, ontology, theology, and human bodily and intellectual (spiritual) development.
In short, Mulla Sadra defines time as an abstraction from instability and motion inherent to the material world, with no actual, independent existence.
In other words, concept of time is a result of perception of change or motion in the world. Time has thus no actual existence. What actually exists is motion, and change/motion is an essential property of matter. Obviously concepts of time such as 'past', 'present' and 'future' don't have any actual referent in the external world. These concepts are rooted in our mind, i.e. how we mentally categorize and conceptually preserve the lasting impact of a constantly changing world on our memory. Hence, when we say a day passes, it actually means some realities no longer exist. They are "past" but the memory of that bygone reality enables us to recall it in our mind and categorize it as 'yesterday', 'past', or other adverbs or adjectives of time.
One important implication of this theory for philosophy of mind is that it points to the extra-natural essence of human mind. If human mind was not immaterial and thus constantly had its substance changed through time (like material forms), then we could have never have a concept of time, because no perception of reality would be preserved to be recalled later, giving us the sense of time. This in turn reveals that mental entities unlike material entities are in essence static and immutable.
Mulla Sadra therefore defines time as the fourth dimension of the material world. The material plane of existence is stretched in 3 dimensions of space and 1 dimension of time. Dimensions both suggest gradual, divisive and imperfect existence of the material beings as opposed to the static, inclusive and whole existence of supernatural beings that are never subject to change and thus time.
Mulla Sadra's theory of time is a part of his greater theory of Substantial Motion which is the main cornerstone of his theory of human bodily and spiritual development among others.
Personally, I think that Aristotle still has the best thing to say about what time is.
Time is the measure of motion.
The root of the idea is that time and motion are interdefined. One is not more primitive than the other. Think about what it is for a ball to roll across the table. At one time it was here, now it is there. What is for the ball to move, it is for it to differ in location with respect to time. What is it for time to pass is simply for change (motion) to occur.
Time is not yet fully understood, a "correct" definition is therefore not possible.
If you understand time as a 4th dimension, then time is what it takes to get a system to change from one state to another. Basically a distance between two of those states.
If you follow the idea of a 6-dimensional universe (or quantum state universe), then time is a chain of possibilities that you just happen to perceive. Your lack of knowledge about other chains of possibilities causes you to think of time as a flow in a singular universe.
If you think like Descartes, then time could be completely fragmented and you'd be jumping from moment to moment through time at total randomness, but if you'd only remember what happened before each moment you are in, you would think of time as continuously even though it is not. Or in reverse: time does not exist: you only perceive this very moment and have false memories about everything that happened before, and cease to exist right after this thought, but you still perceive time as a continuous stream.
It is easy to prove, that time exists independently of men: just take a camera and have it film any scene on its own without anyone around. Time will still continue, as otherwise the camera would not be able to take the pictures. If you however take the idea of Descartes' demon that lets you see whatever it wants you to see, then of course one could argue that the cam did not record anything until that very moment you pressed the play button. I do not know of any experiment that proves that idea wrong.
What however is "proven" by the relativity theory (as in: the theory has been proven so many times that it can be assumed to be correct even for this imaginary situation) is that time is not only perceived differently for every observer, it actually behaves differently. An example from my physics book: take a train that travels at near-light speed. In the center of that train is a switch that switches on both lights at the front and at the rear of the train simultaneously. If that switch is pulled, then an observer inside the train would see both lights to go on simultaneously. An observer outside the train however would see the rear lights switch on first and the front lights switch on later (or even never), as the information (electricity) cannot travel faster than light. It is important to understand that this is not a perceiving problem, it actually happens like that for both people. So basically the state of the world differs depending on where you are when the lights are switched on.
Time is perception of change.
The second hand unwinding, the pattern of night and day, the cycle of life, a raindrop falling to the ground... all of these may be perceived as matter that is interacting with forces of energy - including those of decay upon a material, even subatomic level.
A game of chess, abandoned mid-game with pieces upon a table... this is how a momentary instant in time may be perceived.
One may reference generations, millenia of people before us who lived entire lives - but these too are manifestations of change. The matter of those generations still exist - simply transformed - to take up mineral form - perhaps even incorporated into new life.
Time existed long before the dawn of humanity. Other creatures would have been able to appreciate the changes that went on around them - even if on the basest of levels.
Even the inanimate, while not possessive of the faculties that we consider to be perspective or intelligence, may be said to have a vested interest in their being (A comet passing by a sun, blasting ice and steam from its surface for instance, or a stone fallen into a stream made smooth as it comes to interact with more liquid molecules to become a smoothed pebble)). Retrospective, I know.
Note that this perspective precludes the possibility of time travel - although it does not preclude the possibility of recreating elements of the past (sufficient data, materials and technology provided).
There are several senses of time.
There is physical time, measured by natural periodic oscillations, like by a pendulum, astronomical regularities, or atomic vibrations.
There is phenomenal time, as measured by each person's subjective judgment of time.
There is also what might be called effective time which is measured by changing spatial relations between objects, as is illustrated by changing positions in a game of chess.
As Avestron said above, "A game of chess, abandoned mid-game with pieces upon a table... this is how a momentary instant in time may be perceived." At some other such instant, there are fewer pieces on the board, but black has fewer than white. White has effectively progressed faster than black (known as greater 'tempo').
As one answer suggested, time is changing states. An object without time cannot move; but if the present state changes, the object is free to move.
How are states changed? Perhaps the change is from a force that exists without time, like a gravitational wave (note that the formula for gravity doesn't include time). The wave does not move the object, but changes its state. Then energy can move the object. While this process is not proven, it makes a plausible answer.
"The present is only a line of demarcation between the past and the future; so we cannot rationally say that we care only for the present, as it has no existence apart from the past and the future. It is all one complete whole, the idea of time being merely a condition imposed upon us by the form of our understanding." - Swami Vivekananda
As a point of comparison with this question and its context of philosophy, some people may be interested in the long discussion of time in the context of the intersection between quantum mechanics and the minds of living animals. (Yes, I know, it's peculiar.)
I will address the challenge to give a short definition:
I would argue that time is the direction through space in which composite entropy increases most rapidly, and that the Second Law of Thermodynamics is just a definition.
Our memory is an exothermic chemical process, so ongoing time is just our tracking of the entropy increase. Our notion of time before animals came on the scene is just a projection of that perceived process toward the lowest-entropy state. It does not really exist.
The weakest point of this notion is explaining how our shared notion of time propagates, so that time is not a local phenomenon, but seems to be a consistent and cohesive whole. On the other hand, given General Relativity, we should not get too hung up on the distinction between local and global, as we are just starting to really understand what 'local' really means.
The idea that time passes slower when there are a lot of particles around (i.e. gravity dilates time) seems oddly consistent with Leibniz idea that change is an act of decision, and each monad reflects the changes in the monads it is aware of in deciding how to proceed.
