On the origin of Entropy

With respect to time, there are two classes of evolution : irreversible and reversible. An irreversible process is one that marks a definitive arrow of time, meaning that the process occurs such that the entropy of the system increases as a function of time. On the other hand, a reversible process is a trajectory of constant entropy. Either way, barring overwhelmingly infrequent large fluctuations, entropy of a system cannot decrease with time.

Equilibrium comprises states of constant entropy; more precisely those where entropy is maximized (and hence, time becomes inconsequential). Non-equilibrium processes are always accompanied by an arrow of time. This monotonic trend, or the law of increase of entropy (with time) is a rather intrinsic feature of Nature. Classical Mechanics, which firmly stands on pillars of time-reversal invariance (as Newtonian equations of motion are second order in time) cannot thereby lead to a microscopic understanding of this fundamental nature of the arrow of time. 

Interestingly, the Schroedinger equation in Quantum Mechanics, being first order in time, intrinsically encapsulates a temporal directionality; and therefore reasserts the idea that the hope of a fundamental understanding of the law of increase of entropy resides in a quantum description of Nature.

This anyway is an intuitive approach since a 'true' microscopic picture of any naturally occuring system has to be Quantum Mechanical. The answer to the fundamental, and rather philosophical question of what produces the arrow of time lies therefore in understanding such fundamental systems which explicitly break time-reversal invariance.

Suggested further reading for a technical understanding: Statistical Physics by Landau-Lifshitz, Vol 5, Chapter 1.

An alternative and interesting classical approach towards a better understanding of this directionality in time is through overdamped dynamics, usually effective in modeling the motions of  different biological organisms. The mathematical ingredients hide in the first order nature of the dynamical equations, and the 'persistence' of memory, typically encoded in short-ranged temporal noise correlations. While at present this approach has been quite succesful in providing answers related to the 'effects' of breaking time-reversal symmetry, the queries on the 'origin(s)' of time-reversal invariance, still remain elusive. Maybe, there is no way out other than incorporating quantum ideas for these biological phenomenologies. 

In any case, our understanding of Entropy itself seems to carry an arrow of time on its young shoulders, and a lot of disorder is on its way.