What does "entropy" really mean? I'm currently taking a course of introduction to biophysics in college and we currently covered a subject called entropy (S). We did some maths based on the Carnot cycle and came up with the following formula for this state function:
$$\partial S = \partial Q /\partial T $$
However the professor later stated that this function we call entropy measures the ability of a system to do work. How exactly does $\partial Q /\partial T$ gives us that information??
I'm confused, can someone please explain that to me in layman's terms? I'm not a physicist.
 A: 
I'm confused, can someone please explain that to me in layman's terms?
I'm not a physicist.

The mathematical aspects of entropy are probably beyond layman's terms. Insofar as the link provided by @Jonas, although it's good, i'm not sure how helpful it would be to a layman.
Probably the most intuitive explanation of the second law and entropy that most folks can understand is rooted in everyday experiences. It has to do with the natural direction of processes. For example we all know

*

*Heat only flows naturally (without help) from hot to cold objects and


*Rivers always flow down-hill
And one that you, a biophysics student can relate to:


*People always grow old

These are all examples of what we call irreversible processes (they don't naturally reverse). For each process, entropy is generated. But all of these processes obey the first law of thermodynamics, which is basically conservation of energy.
For example, if heat flowed naturally from a cold object to hot object, the heat lost by the cold object would equal the heat gained by the hot object and energy would be conserved. But we never observe this to happen.
If water flowed naturally up hill it would lose kinetic energy and gain gravitational potential energy and energy would be conserved. But we never observe this to happen.
As far as the third example is concerned, ask your teacher how the second law applies to it.
Scientists (Notably Clausius and Carnot) realized that there must be another law and property governing the direction of processes in addition to the law of conservation of energy. Enter the second law and entropy.
Now as regards to the definition of entropy, the proper form of the equation is
$$dS=\frac{\delta Q_{rev}}{T}$$
Which says that a differential change in the property called entropy equals a reversible transfer of heat $\delta Q$ divided by the temperature at which the transfer occurs.
Entropy is generated when the heat transfer is irreversible, i.e., when it occurs over a finite temperature difference. This entropy generation may result in the lost opportunity to do work. A mathematical explanation is needed to show how. Your teacher should be able to help.
Hope it helps.
A: A somewhat similar question answered by me may be helpful in this case (please read that up untill the equations, at least, and continue here).  Concluding comments (to the aforementioned answer) -  Loosely speaking from equation (1), the entropy of a system is proportional to the number of accessible quantum states (this is what entropy is!) → more the amount of (co-ordinate/momentum) space available, higher the entropy, higher the disorder (because there will be more choices available for a particular particle to settle into - so, "order" does not mean a well-arranged/ordered outer appearance but refers to the number of accessible quantum states in  & , neither it is a relative term).
Thus, a disordered state is simply more probable because by the very definition above there are more states available to be occupied - for eg., this essentially means a particular particle has higher probability to occupy that state which has higher frequency of occurence = eg. if a system has more spin-up (|↑⟩) states available than spin down states (|↓⟩), then the particle has higher probabilty to be spin-up rather than spin-down.
