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Non-equilibrium systems driven away from equilibrium develop sustained currents and dissipation. Response to weak external fields is still given by the fluctuation-dissipation theorem when the system remains close to equilibrium (linear response). Far from equilibrium, novel phenomena emerge: bistability, spatiotemporal chaos, self-organization. The theory connects microscopic dynamics to macroscopic irreversibility and transport.
Equilibrium statistical mechanics — Boltzmann factors, partition functions, thermodynamic potentials — describes systems that have relaxed and are no longer changing. Most systems in nature are not in this state. A current-carrying wire, a living cell, a weather system: these are maintained away from equilibrium by external drives (voltage, chemical gradients, solar energy). Non-equilibrium statistical mechanics asks how to describe and predict such systems. From your study of the H-theorem and reversibility, you know that microscopic dynamics are time-reversible but macroscopic entropy increases. Non-equilibrium theory makes this precise: irreversibility is not an approximation but a feature of how macroscopic descriptions are constructed.
The conceptually simplest regime is linear response: the system is close to equilibrium, and the external drive is weak. Here, the response to a perturbation (the induced current, magnetization, or polarization) is proportional to the drive. The proportionality constants — conductivity, susceptibility, diffusivity — are called transport coefficients. The deep result, which your prerequisite on the fluctuation-dissipation theorem established, is that these coefficients are determined by the equilibrium fluctuations of the system in the absence of any drive. You do not need to apply a voltage to measure conductivity — in principle, you can read it off from how current fluctuations decay at equilibrium. This unifies the description of dissipation and fluctuations.
Moving further from equilibrium, linear response breaks down. The relationship between currents and forces becomes nonlinear, and the system can develop multiple stable states (bistability) — systems that can sit in either of two macroscopic configurations and switch between them. Driving still further, temporal periodicity (limit cycles), spatiotemporal patterns (like Turing patterns in chemical reactions or convection rolls in heated fluids), and ultimately chaos can emerge. These phenomena are not accessible from equilibrium statistical mechanics and require a separate framework — often built around nonlinear differential equations for macroscopic order parameters.
The unifying thread is entropy production. In equilibrium, entropy production is zero. Close to equilibrium, Onsager's minimum entropy production principle constrains the steady state. Far from equilibrium, no such minimum principle holds, and the system can organize into low-entropy structures maintained by continuous dissipation — what Prigogine called dissipative structures. This is the statistical mechanical basis for understanding why living organisms, which are quintessentially non-equilibrium, are locally ordered while globally increasing entropy: they are sustained dissipative structures, maintained by a continuous flow of free energy from food (or sunlight) to heat.