We call a system “deterministic” if we believe that mechanics can tell us exactly what this system will be doing at some future point. It’s like how we’re certain that a ball will roll down a hill, or that a particle will descend into a potential well, to use two equally simple examples. There are two kinds of mechanics, classical and quantum. There are some differences between them, and you might have heard alot about these differences. What you might not have heard as much about is the similarities between them, and furthermore the fact that neither discipline constitutes, by itself, the whole of physics. Thermodynamics developed as a distinct discipline apart from either kind of mechanics and these disciplines were not unified until Boltzmann laid the foundations for statistical mechanics, at the end of the 19th century. Both classical and quantum mechanics, however, deal with the mechanisms which govern the motion of individual objects, whether bodies or wave functions. Furthermore, the fundamental rules of mechanics are equations which are time-reversible and must be derived by abstracting away from the multiplicity of reality and imagining the behavior of one single thing, a body being subjected to a force or a particle in a box which you see is actually a standing wave--but not until you think about what its like if its the only particle in the box.
Abstracting away from reality to the behavior of a single object has been one of the great achievements of modern science, and no discipline has been more successful and had more predictive force than mechanics. But we know, thanks to Poincare and Boltzmann and others who came later in the 20th century, that the behavior of many things is not at all like the behavior of one thing. Crucially, the behavior of many things can display a property we call irreversibility. One thing by itself can’t do anything irreversible. But an ensemble can, and once irreversibility enters the world, so does history.
An irreversible process, such as a smoking cigarrete, can only proceed one direction in time. This directedness is not deducible from the behavior of an individual particle in the smoke.