The laws of nature determine how systems evolve in time. In other words, if one had a complete description of a system at some initial moment, its future would in principle be fixed.
This is the basic idea of classical determinism.
Laws of nature act like rules that convert an input into a unique output. If the input is fully specified, the outcome is fully specified as well. Any statistical aspect enters not because the laws themselves are uncertain, but because our knowledge of the initial state is incomplete.
A simple example is a stone tossed vertically upward. If we know its initial speed precisely, then we can calculate how long it will take to return to the ground. In reality, we usually do not know that speed exactly. We may only know it within some range, or assign a probability distribution to it. The uncertainty in the initial condition is then translated by the laws into an uncertainty in the outcome, in this case the return time.
This is also why chaotic systems are so hard to predict. In such systems, very small differences in the initial conditions can produce very large differences later on. The laws are still deterministic. What changes is that even a tiny initial uncertainty can grow so quickly that practical prediction soon becomes useless. This is why a weather forecast can be accurate in the near future and yet become unreliable further ahead.
This issue becomes increasingly severe with increasing complexity of systems. In principle, their future is still fixed by their initial state and by the laws governing them. In practice, however, the calculation becomes overwhelmingly difficult. The number of relevant variables becomes too large, the interactions too entangled, and the required precision too demanding. One then tries to predict only part of the full picture, treating outside influences approximately or ignoring them altogether. The result can still be useful, but it is no longer exact.
This is an important distinction. Classical determinism is not the claim that we can actually compute the future. It is the claim that the future is fixed, whether or not we are able to calculate it.
Taken seriously, this idea becomes mind-bending when applied to the universe as a whole. There was a time in the history of the universe when it was in a far simpler state than it is today: a hot soup of particles, long before stars, planets, and living beings appeared. Even though we do not know the exact positions and velocities of every particle in that early plasma, classical determinism says that this does not matter in principle. What matters is that the full state existed, and that, together with the laws of nature, it fixed everything that followed.
From that early cosmic soup, with all its tiny inhomogeneities, the later universe would unfold in a unique way. The formation of galaxies, stars, planets, weather systems, biological life, and human societies would all be, in principle, consequences of that earlier state.
This is the real force of classical determinism. It says that the apparent unpredictability of the world is not necessarily a sign that the world itself is open at the fundamental level. It may simply reflect the enormous complexity that has emerged as the universe evolved. The future may be practically unknowable while still being predetermined in principle.
That possibility has fascinated people for a very long time. The reason is that, if classical physics is taken at face value, then the universe does not merely evolve lawfully. It evolves toward its inevitable outcome.
Whether the universe is still deterministic in this sense once quantum physics enters the picture is a separate question.
See also: How did structure form in the universe?