Time is one of the most fascinating concepts in physics.
It is sometimes presented as if physics still does not know what time really is, whether it truly flows, or why it has a direction. That picture is misleading. At every level at which we currently describe nature, whether classical or quantum, time enters as a well-defined part of the laws. There has never been a basic uncertainty in physics about the physical meaning of time evolution. It is true that time is sometimes said to become peculiar in quantum gravity, but even there the situation is not as mysterious as it is often made to sound. For the moment, it is better to set those peculiarities aside and focus on the broader concept.
A different question is why living beings such as ourselves experience time the way they do. That is a question about perception, memory, and the structure of conscious organisms. Here we will leave that aside as well and focus on the role time plays in the fundamental description of nature.
A physical theory is defined by two ingredients. First, it tells us what states the universe can be in. That list includes empty space, a single-particle state, a cup of tea, five pink elephants dancing to music, and a universe filled with galaxies. Second, it gives laws that relate these states to one another under a continuous flow. That flow is what physics calls time.
To understand what this means, it is better not to begin with clocks. A clock is just one physical system among many others. Time is the parameter with respect to which the laws of nature generate change in every physical system.
As a simple example, imagine an electron and a positron initially at rest. Besides that electron-positron state, there are other physical states that contain only light moving away in different directions. The laws of nature tell us how the initial state evolves as the time parameter changes. The two particles attract one another, move, radiate light in the process, and may eventually annihilate into a state containing only light. More precisely, the state evolves into a quantum superposition involving electron-positron states, which may already contain light radiated during the evolution, and states containing only light.
The same is true of us. If we sit at a desk looking at a clock, then as time increases the hands of the clock move, our heart beats, our eyes blink, our cells divide, and thoughts succeed one another. The point is not merely that many things happen together. The point is that they are all governed by the same continuous parameter entering the laws of nature. In that sense, there is one fundamental flow according to which all physical systems evolve.
This is what physics says about time. Most importantly, physics often tells us how things work rather than why. The laws of physics have no difficulty with time as a parameter of evolution, a fundamental flow of nature that transforms one physical state into another.
Some of the more exotic effects associated with relativity, such as time dilation or the role of curved spacetime, do not undermine this basic picture. They modify how different physical systems track time relative to one another, but they do not abolish time as part of the fundamental description. To simplify the discussion, one can choose a single time parameter and describe the experience of each observer as a function of it. Even in gravity, and even after quantization, one still has lawful flow of time.
Hermann Hesse touched upon something close to this in Siddhartha, where the ferryman speaks of the river as being everywhere at once: at its source and at its mouth, at the waterfall, at the ferry, on the rapids, in the sea. The fact that one has not yet reached the end does not make it less real. Siddhartha then sees his own life in the same way, with childhood, adulthood, and old age separated not by reality, but only by shadows. Here Hesse captured certain aspects of time extremely well, including features that echo some of the more peculiar ways it appears in modern fundamental physics.
See also: What is the language of the universe?