Thermodynamics. The second law of thermodynamics states that entropy always increases in an isolated system. This is taken as a fundamental postulate---we simply accept this statement as a fact regarding how the world works, and our justification is that no experiment has ever shown the second law incorrect. In the framework of macroscopic thermodynamics, there is no deeper principle from which we can hope to derive the second law.
Statistical mechanics. Statistical mechanics brings a microscopic perspective into thermodynamics. We talk of macrostates---macroscopic states of our system, characterized by (macroscopic) observables like pressure, temperature, volume, etc.---which may be realized by a number of microstates---a complete microscopic description of our system, consisting of the positions and momenta of each particle. Many different microstates can correspond to the same macrostate, since a macrostate only deals with macroscopic properties.
In statistical mechanics we can go one level deeper, taking as a fundamental postulate the principle of equal a priori probabilities, the idea that the system has an equal probability of being in any given microstate. This implies that the most likely macrostate for the system is that with the greatest number of corresponding microstates. At this point, knowing also from statistical mechanics that the entropy is a monotonically increasing function of the number of microstates (specifically the logarithm of the number of microstates in an isolated system, but the functional form is unnecessary here), we can motivate the second law: the entropy of a macrostate of an isolated system is a measure of the number of its corresponding microstates; the greater this number, the more likely this macrostate will be observed. The equilibrium state of a system is simply its most probable macrostate.
Example. (Microscopic diffusion.) Let us consider a microscopic model of diffusion: a $10$-by-$10$ grid of particles, $50$ red and $50$ blue. The particles are indistinguishable except by color. In addition, we will consider two sets of macroscopic states, 'ordered' and 'disordered'. Take as initial state the configuration with the two groups of particles separated by color into two equal halves by a horizontal boundary, the blue particles on the top and the red on the bottom. The diffusion process is akin to removing this artificial boundary.
There is one initial ordered microstate and $100!/(50!)^2$ final microstates. (Note that indistinguishable microstates should be considered the same microstate.) The entropy of our system has increased, because we have relaxed a constraint and allowed more microstates into our system. Most of these final states look disordered. Diffusion is therefore an entropically favorable process that brings an ordered system into a disordered one.
Additional remark. I find statements like "entropy drives the system towards a particular state" to be somewhat misleading, because they imply that entropy behaves like a force. But entropy, as we've seen, is really just statistics.