What exactly is an entropic force? In general the entropic force is defined as

$\vec{F}(\vec{X}_0) = T \vec{\nabla} S(\vec{X})\vert_{\vec{X}=\vec{X}_0}$,

where $T$ is the temperature of the system and $S(\vec{X})$ is the entropy associated to the macrostate $\vec{X}$ and $\vec{X}_0$ is the present macrostate.
I think, that I understand the argument, that a systems tends to evolve in such a manner, that the entropy increases. However I am not sure how to understand this kind of force. Is the entropic force some kind of average force?
 A: Even if we ignore fundamental-force interactions (that would produce internal stresses or gravitational tugs or electrostatic repulsion or magnetic attraction, for example) and assume nobody comes over to push on our system, the system's particles can still receive energy from their surrounding environment simply from the fact that that environment is above absolute zero. This is the reason that the temperature $T$ appears as a linear factor in entropic force calculations.
Nonzero temperatures (which are all temperatures in practice) provide the energy for the system to fluctuate and spontaneously explore various configurations, of which we observe the one most likely to occur.
A classic example is the ideal gas, whose pressure arises not from any repulsion but from the tendency of noninteracting particles at nonzero temperature to explore the available volume. Another is the ideal elastomer, which retracts when you pull on it not because of any attraction but because the now-unkinked, uncoiled molecule naturally tends to contract again as it explores geometric configurations. (Think of an originally straight chain on a vibrating table—after being shaken around a bit, the endpoints have inevitably moved closer.)
It might be useful to consider a thermodynamic model of the latter to see the division between enthalpic forces and entropic forces. We can add energy to a strip of elastomer (such as uncrosslinked neoprene) by heating it, pressurizing it, or stretching it, among other ways. We can write the fundamental relation in differential form as $$dU=T\,dS+f\,dl,\tag{1}$$ where $U$ is the internal energy, $T$ is the temperature, $S$ is the entropy, $f$ is the force, and $l$ is the length. (The typical pressure–volume term doesn't appear because volumes don't change much for condensed matter.) Since we're operating slowly, let's assume constant temperature as we take the derivative with respect to $l$:
$$f=\left(\frac{\partial U}{\partial l}\right)_T-T\left(\frac{\partial S}{\partial l}\right)_T\tag{2}.$$
Materials such as metals, ceramics, and strongly crosslinked polymers obtain their stiffness from the $(\partial U/\partial l)_T$ term, as their entropy doesn't increase by much upon an elastic strain of, say, 0.1%, which is nearly all that their bonds can sustain. Elastomers are different. Without crosslinks, there is little preventing their long, kinked molecules from extending; thus, they gain their stiffness almost entirely from the $-T(\partial S/\partial l)_T$ term. In our idealization of the extreme case, the ideal elastomer gains no potential energy when stretched—its stiffness is entirely entropic.
You could think of the entropic force as an average force in the sense that the system must explore these different configurations over finite time. You could also think of it as a temperature-driven force (since that linear factor of $T$ always appears) or a force arising from the thermal energy of the surrounding heat bath.
