Gibbs free energy intuition What is Gibbs free energy? As my book explains:

Gibbs energy is the energy of a system available for work.

So, what does it want to tell? Why is it free? Energy means ability to do work. What is so special about this energy? Can anyone simply explain? 
I just want a math-free intuition.
 A: Gibbs free energy is a thermodynamic potential that measures the "usefulness" or process-initiating work obtainable from a thermodynamic system at a constant temperature and pressure (isothermal, isobaric). Just as in mechanics, where potential energy is defined as capacity to do work, similarly, different potentials have different meanings.  The Gibbs free energy is the maximum amount of non-expansion work that can be extracted from a closed system; this maximum can be attained only in a completely reversible process. When a system changes from a well-defined initial state to a well-defined final state, the Gibbs free energy $ΔG$ equals the work exchanged by the system with its surroundings, minus the work of the pressure forces, during a reversible transformation of the system from the same initial state to the same final state.
Gibbs free energy is also the chemical potential that is minimized when a system reaches equilibrium at constant pressure and temperature. Its derivative, with respect to the reaction coordinate of the system, vanishes at the equilibrium point. As such, it is a convenient criterion of spontaneity for processes with constant pressure and temperature.
A: At constant temperature and pressure, Gibb's free energy is the criteria that determine equilibrium for a system (*). More precisely put, the system will keep changing its state until it falls into a state where $ \Delta G = 0$ for the constraint conditions mentioned above. Similar to the principles of potential, the reaction is driven to drop the Gibbs free energy to zero.
Naturally, we would see that

*

*$ \Delta G > 0$ non-spontaneous processes

*$ \Delta G< 0 $ for a spontaneous process

*$ \Delta G = 0 $ for an equilibrium process

Different constraints have different conditions for equilibrium, however, for the one mentioned above, Gibb's is the one.
The reason for it being called the maximum work is that the expression was derived with considerations of the reversible process being done (**)

Proving (*) and (**):
From the second law of thermodynamics, it is a condition that for any thermodynamic process must go in a direction of increasing entropy. Writing this as a mathematical expression:
$$ \Delta S_{\text{universe}} = \Delta S_{\text{sys}} + \Delta S_{\text{surr}} \geq 0$$
Now,
$$ \Delta S_{\text{sys}} + \Delta S_{\text{surr}} \geq 0 $$
or,
$$ \Delta S_{\text{sys}} \geq - \Delta S_{\text{surr}}$$
Now any spontaneous process follows the above equation as it is an implicit statement about the second law of thermodynamics. Secondly, I'll ditch the suffix of 'sys' and just call the entropy of the system as 'S'. Finally, due to the definition of entropy and process being defined as adiabatic (***),
$$ \Delta S_{\text{surr}} = \frac{dq_{\text{surr},\text{rev}} }{T_{\text{surr}} }  = \frac{-dq_{\text{sys,rev}} }{T_{\text{surr}} } $$
Now, we plug this relation back into the restatement of the second law in differential form,
$$ dS \geq \frac{dq_{\text{sys,rev}} }{ T_{\text{surr}} }$$
Applying the first law of thermodynamics onto the system,
$$ \mathrm{dU} = \mathrm{đq_{\text{sys,rev}}} + \mathrm{dW} = \mathrm{dq_{\text{sys,rev}}} - \mathrm{P_{\text{ext}} dV}$$
and with some algebra,
$$ dU + P_{\text{ext}} dV = dq_{\text{sys,rev}}$$
Plugging this into the expression for the entropy of the system,
$$ \Delta S =  \frac{ \mathrm{dU} + P_{\text{ext}} \mathrm{dV}}{T_{\text{surr}}}$$
Rearranging and plugging into inequality, we get the general criteria for spontaneous change as:
$$ \mathrm{dU} + P_{\text{ext}} \mathrm{dV} - T_{\text{surr}} \mathrm{dS} \leq  0 $$
Now, under the 'experimentally useful (****)' conditions of constant temperature and external pressure,  we can write the first two terms as enthalpy i.e: $ \mathrm{dU} + P_{\text{ext}} \mathrm{dV}$. Hence the inequality becomes,
$$ \mathrm{dH} - T_{\text{surr}} \mathrm{dS} \leq  0 $$
Atlas, the definition of Gibb's free energy comes out naturally! We precisely define the expression on the left-hand side of the expression as Gibb's free energy. Now the change in this quantity being negative exactly the criteria for spontaneous change under constant temperature and pressure.
$$ \Delta G \leq 0$$
So, what does this mean? The system will keep changing state spontaneously until finally it gets stuck in a state where Gibb's free energy change between some states nearby in its state-space is zero.
Regarding the 'maximum work' thing, of course, you would only get the maximum work output when you have a reversible process This was one of the assumptions in deriving the result of Gibb's free energy as a criterion of spontaneous change in the constraints mentioned before in the write-up. The opposite idea is that if we did the process is irreversibly, then we'd have distortion of the system such as vicious stresses which would eat up the energy from the non-expansion work of the system.

Footnotes:
(***): The system of surroundings + universe is adiabatic as there is nowhere other than the universe where heat can go. Due to this, the heat into the system is the same as heat lost by the universe.
(****): Most chemical reactions are done outdoors where you can approximate pressure and temperature as approximately a constant. The atmospheric pressure and temperature of the atmosphere do not fluctuate much when you run a process in the open air.
It is free in the sense that it is the measure of energy that we could freely get out of a system that is not in equilibrium.

References:
MIT OCW notes
Second Law of Thermodynamics
Discussion of entropy expressions: Refer page-120 of this book
Lecture on this specific topic by MIT OCW
A pop-science segment from a movie that made me think entropy is cool: Mr. Nobody
A: In the simplest sense of it, the Free Energy is the heat of the system minus the compulsory heat loss due to entropy. So, in short, it is the amount of "energy" left over in the system, after we consider losses due to entropy. So basically, some amount of heat is wasted, and the remaining amount is useful. And, this remaining amount is the Gibbs Free Energy. 
$G=H-TS$, here $H$ is the amount of energy contained in the system, and $TS$ is the amount that will inevitably be wasted. That's how it makes sense to me anyway.
A: There are two forces influencing the spontaneity of a reaction:
(1) The tendency of a system to attain a state of minimum energy and maximum order or stability.
(2) The tendency of a system to attain a state of maximum energy and minimum order or entropy.
If a system attains maximum stability, it attains minimum entropy; if it attains maximum entropy, it attains minimum stability. Therefore, the two always oppose each other. If we want to predict the spontaneity of a reaction, we must take into account the effect of both forces. But how? 
Since we need both criteria (which are in opposition) in determining whether a given reaction will be spontaneous or not, an equation is needed that takes into account both criteria. And, that equation is the Gibbs Free energy equation. The $\Delta H$ term represents the first criterion (stability) and the $T\Delta S$ term represents the second criterion (entropy). The minus sign before $T\Delta S$ means that the two criteria are in opposition. 
Note that equilibrium is achieved when stability ($\Delta H$) and entropy ($T\Delta S$) work out a compromise such that both are satisfied, like when neither side can succeed at defeating the other in a game of tug-of-war. Then $$\Delta H=T\Delta S \\\\\\\\ \implies \boxed {\Delta G=0} \quad (\text{equilibrium condition})$$.
Please feel free to edit the answer and improve it.
A: First, you have system with some energy, named $U$ by physicists. You think you have all the information you need to characterize the system but then some guy comes near and says:
"Whoa, that's bad, the volume of your system can change."
You say: 
"No problem, we just add here $pV$. Our new energy is $H=U+pV$."
"But hey," they say, "your temperature can change by external heat, you have to count that also."
"No problem, we subtract $TS$ from our energy, and rename it. $G=U+pV-TS$"
There you are.
People name it "available" energy, because if your system increases in size, it creates work: $V$ increases, so $U$ decreases. But energy is always bounded from below. When you drain everything you can, your system goes to $G=0$.
You cannot drain all $U$ because you would need an external system with $T=0$.
A: Sanaris's answer is a great, succinct list of what each term in the free energy expression stands for: I'm going to concentrate on the $T\,S$ term (which you likely find the most mysterious) and hopefully give a little more physical intuition. Let's also think of a chemical or other reaction, so that we can concretely talk about a system changing and thus making some of its internal energy $H=U+p\,V$ available for work.
The $T S$ term arises roughly from the energy that is needed to "fill up" the rotational, vibrational, translational and otherwise distractional thermal energies of the constituents of a system. Simplistically, you can kind of think of its being related to the idea that you must use some of the energy released to make sure that the reaction products are "filled up" with heat so that they are at the same temperature as the reactants. So the $T S$ term is related to, but not the same as, the notion of heat capacity: let's look at this a bit further. 
Why can't we get at all the energy $\Delta H$? Well, actually we can in certain contrived circumstances. It's just that these circumstances are not useful for calculating how much energy we can practically get to. Let's think of the burning of hydrogen:
$$\rm H_2+\frac{1}{2} O_2\to H_2O;\quad\Delta H \approx 143{\rm kJ\,mol^{-1}}\tag{1}$$
This is a highly exothermic one, and also one of the reactions of choice if you want to throw three astronauts, fifty tons of kit and about a twentieth of the most advanced-economy-in-the-world’s-1960s GDP at the Moon. 
The thing about one mole of $H_2O$ is that it can soak up less heat than the mole of $H_2$ and half a mole of $O_2$; naively this would seem to say that we can get more heat than the enthalpy change $\Delta H$, but this is not so. We imagine a thought experiment, where we have  a gigantic array of enormous heat pads (all individually like an equilibrium “outside world") representing all temperatures between absolute zero and  $T_0$ with a very fine temperature spacing $\Delta T$ between them. On my darker days I find myself imagining an experimental kit that looks eerily like a huge pallet on wheels of mortuary shelves, sliding in and out as needed by the experimenter! We bring the reactants into contact with the first heat pad, which is at a temperature $T_1 = T_0 - \Delta T$ a teeny-tiny bit cooler than $T_0$  thus reversibly drawing some heat $\Delta Q(T_1)$  off into the heat pad. Next, we bring the reactants into contact with the second heat pad at temperature $T_2 = T_0 - 2\,\Delta T$, thus reversibly drawing heat $\Delta Q(T_2)$ off into that heat pad. We keep cooling then shifting to the next lowest heat pad until we have visited all the heat pads and thus sucked all the heat off into our heat pads: see my sketch below:

Now the reactants are at absolute zero. There is no heat needed to "fill them up" to their temperature, so we can extract all the enthalpy $\Delta H$ from the reaction as useful work. Let's imagine we can put this work aside in some ultrafuturistic perfect capacitor, or some such lossless storage for the time being.
Now we must heat our reaction products back up to standard temperature, so that we know what we can get out of our reaction if the conditions do not change. So, we simply do the reverse, as sketched below:

Notice that I said that $H_2O$ soaks up less heat than the reactants. This means that, as we heat the products back up to standard temperature, we take from the heat pads less heat in warming up the water than we put into them in cooling the reactants down. 
So far, so good. We have gotten all the work $\Delta H$ out into our ultracapacitor without losing any! And we're back to our beginning conditions, or so it seems! What's the catch?
The experimental apparatus that let us pull this trick off is NOT back at its beginning state. We have left heat in the heat pads. We have thus degraded them: they have warmed up ever so slightly and so cannot be used indefinitely to repeatedly do this trick. If we tried to do the trick too many times, eventually the heat pads would be at ambient temperature and would not work any more.
So we haven’t reckoned the free energy at the standard conditions, rather we have simply calculated the free energy $\Delta H$ available in the presence of our unrealistic heat sink array. To restore the system to its beginning state and calculate what work we could get if there were no heat sink array here, we must take away the nett heat flows we added to all the heat pads and send them into the outside World at temperature $T_0$. This is the only "fair" measure, because it represents something that we could do with arbitrarily large quantities of reactants.
But the outside World at $T_0$ is warmer than any of the heat pads, so of course this heat transfer can’t happen spontaneously, simply by dent of Carnot’s statement of the second law! 
We must therefore bring in a reversible heat pump and use some of our work $\Delta H$ to pump this heat into the outside world to restore standard conditions: we would connect an ideal reversible heat pump to each of the heat pads in turn and restore them to their beginning conditions, as sketched below:

This part of the work that we use to run the heat pump and restore all the heat pads, if you do all the maths, is exactly the $T\,\Delta S$ term.
The above is a mechanism whereby the following statement in Jabirali's Answer holds:

Processes that increase the Gibbs free energy can be shown to increase the entropy of the system plus its surroundings, and will therefore be prevented by the second law of thermodynamics. 

The nice thing about the above is that it is a great way to look at endothermic reactions. In an endothermic reaction, we imagine having an energy bank that we can borrow from temporarily. After we have brought the products back up to temperature, we find we have both borrowed $-\Delta H$ from the energy bank and put less heat back into the heat pads than we took from them. So heat can now flow spontaneously from the environment to the heat pads to restore their beginning state, because the heat pads are all at a lower temperature than the environment. As this heat flows, we can use a reversible heat engine to extract work from the heat flowing down the gradient. This work is, again, $-T\,\Delta S$, which is a positive work gotten from the heat flowing down the temperature gradient. The  $-T\,\Delta S$ can be so positive that we can pay back the $\Delta H$ we borrowed and have some left over. If so, we have an endothermic reaction, and a nett free energy: this energy coming from the heat flowing spontaneously inwards from the environment to fill the higher entropy products (higher than the entropy of the reactants).
Take heed that, in the above, I have implicitly assumed the Nernst Heat Postulate -the not quite correct third law of thermodynamics - see my answer here for more details. For the present discussion, this approximate law is well good enough.
A: The answers provided by other users are very good. But I put this answer as completing them (albeit small). So, the other users can see different viewpoints in this page.
In real life, some processes take place at constant pressure and also at constant temperature. Gibbs Free Energy ($G$) is simply a (Legendre) transformation of energy ($G=E+PV-TS$) in order to construct a state function in terms of pressure and temperature (as variables). Thermodynamically, it is just a function that is used to calculate if a reaction is spontaneous or not. In these cases, the process would be spontaneous if the change of Gibbs energy is negative ($\Delta G < 0$), meaning that the initial state has more free energy than the final state. For example, the familiar process of rusting of iron is an example of a spontaneous reaction in everyday life (No external energy is required to start the reaction!). Also, the metabolism in our body is at constant pressure and temperature, so the definition of Gibbs energy is useful in human body as well. In our body, $\Delta G <0$ means energy is released during the reaction, e.g., glucose reacts with oxygen during glycolysis to release Gibbs energy, which means energy is released in our body.
See the video in this link for obtaining more insight about Gibbs free energy in alive organism such as our body.
Technical definition of $G$: (from the elegant book of Schroeder) If the system is in an environment with constant pressure ($P$) and constant temperature ($T$), then the work
you need to do to create it (the system) from nothing, or the work you can recover when you destroy it (the system) completely, is given by the Gibbs free energy, i.e., $G=E+PV-TS$.
Mathematically, the Gibbs free energy is a new representation of energy by implementing the Legendre transformation as $G=E+PV-TS$. So, thermodynamically, only the boundary conditions of the system have changed and for this reason, microscopically, it can be extracted by use of the definition of partition function as well, which depends on the degrees of freedom of the system (All the state Functions can be derived from the the Partition Function which is supported by the microscopic viewpoint). To make it more clear, it can be extracted from the partition function of the system under study, $Z = {\rm{Tr}}\left( {{e^{ - \beta \hat H}}} \right)$, with appropriate boundary conditions (which specify the ensemble with $P$ and $T$ as the constants of the ensemble) and, in this way, the Gibbs free energy is found as
$$G =  - \beta \left[ {\ln Z - \left( {\frac{{\partial \ln Z}}{{\partial \ln V}}} \right)} \right],
$$
where $\beta  = \frac{1}{{{k_B}T}}$. I should emphasize that the definition of Gibbs free energy is not as simple as internal energy with kinetic and potential terms. In fact, it makes sense more at a macroscopic, thermodynamic level instead of a microscopic level.
A: Short answer: Gibbs free energy $G = U + PV - TS$ combines internal energy $U$, pressure $P$, volume $V$, temperature $T$, and entropy $S$ into a single quantity that measures spontaneity. With that I mean that processes lowering the Gibbs free energy of your system will spontaneously occur, and equilibrium is reached when the Gibbs free energy reaches the lowest possible value.
Processes that increase the Gibbs free energy can be shown to decrease the entropy of the system plus its surroundings, and therefore will be prevented by the second law of thermodynamics. That's why it measures useful energy - your system may contain more energy, but entropy considerations will prevent you from spending it. 
