What is the point of the four different thermodynamic potentials? Why are there four different thermodynamic potentials? What is the point? Nothing new is defined, just a reshuffling? Could someone help me in showing this? Does it relate to doing experiments?
If I understand correctly you swap intensive and extensive variables (e.g. for Helmholtz $S \rightarrow T$)
 A: An an example, imagine that you have an insulated cylinder with a locked piston which is partially filled with liquid H$_2$O.  Some amount of the liquid will evaporate and form an H$_2$O vapor atmosphere which fills the remainder of the container.  In equilibrium, how much of the H$_2$O will be in the vapor phase?
The principle of maximum entropy states that if we obtain a functional relation $S=S(U,x)$ (where $x$ is e.g. the number of moles of H$_2$O in the vapor phase), then the equilibrium value for $x$ is obtained by holding $U$ fixed and maximizing $S$ with respect to $x$.
This principle can be inverted to yield the principle of minimum energy, which states that if we obtain an equation of state $U=U(S,x)$, then the equilibrium value $x$ is obtained by holding $S$ fixed and minimizing $U$ with respect to $x$.
These approaches turn out to be equivalent, so which one you use depends on which equation of state is more convenient to model.

On the other hand, what if the cylinder is not insulated, but instead is allowed to exchange energy with a heat reservoir at temperature $T$? In this case, the equilibrium state is found not by maximizing the entropy of our system by itself, but rather by maximizing the entropy of the system and the reservoir.  This is terribly inconvenient, because it seems to require us to come up with an equation of state involving the reservoir as well.
As it turns out, however, that is not necessary. The only information we need to know about the reservoir turns out to be its temperature; otherwise we are completely free to restrict our attention to our system all by itself.  However, the quantity which is minimized in equilibrium is no longer our system's internal energy $U$, but rather its Helmholtz energy $F := U - TS$.  To obtain our equilibrium state, we hold $T$ fixed and minimize $F$ with respect to $x$.
What if the cylinder remains insulated, but we unlock it and expose it to some ambient pressure $p$ so that the volume can change?  In an abstract sense, this is like putting our system in contact with a volume reservoir rather than an energy reservoir.  Once again, it turns out that if we want to focus on our system then we are free to do so, but once again the internal energy of the system is not the quantity which is minimized in equilibrium. Instead, we should consider its enthalpy $H:= U+ pV$, which we minimize while holding $p$ fixed.
In exactly the same way, if the cylinder can exchange both energy and volume with its environment then the correct quantity to minimize is the Gibbs energy $G := U - TS + pV$, which we minimize while holding both $T$ and $p$ fixed.

So in summary, the motivation for formulating different thermodynamic potentials is that we would like to have some property of our system alone which we can extremize in order to find the equilibrium state of our system. If the extensive parameters ($V,N,$ etc) are all held fixed, then the equilibrium state is obtained by minimizing $U$ at fixed $S$.  However, if the system is allowed to exchange energy, volume, particles, etc. with its environment, then this will not yield the correct result; instead, we will need to minimize the associated thermodynamic potential which is relevant to our imagined scenario.
A: It's mostly about the conditions that must be satisfied to reach equilibrium. Given a specific transformation, you build the appropriate potential that will be minimized when equilibrium is reached.
For example, for a transformation with constant temperature and pression, it's the Gibbs free energy that is the appropriate potential to determine the final equilibrium state.
There are, in fact, an infinite number of potentials. But in "basic" thermodynamics with the usual state parameters (temperature, volume, pressure, entropy...) there is only a small number of different potentials.
Start with internal energy $U$, which is a function of entropy $S$ and volume $V$ (and also numbers of mole, but that won't change anything here):
$$dU=T\,dS-P\,dV+\dots$$
This function will be minized ($dU=0$) after a transformation with $P$ and $V$ constant.
From there you can derive enthalpy $H=U+PV$ which is a function of pressure and entropy:
$$dH=T\,dS+V\,dP+\dots$$
$H$ is the appropriate potential for a transformation with $P$ and $S$ constant: finding its minimum will determine the state of equilibrium.
From there you can derive Gibbs free energy $G=H-TS$, which is a function of pressure and temperature:
$$dG=-S\,dT+V\,dP+\dots$$
It's a very important potential, since transformations with $P$ and $T$ constant are common, for example in chemistry (system in contact with the atmosphere).
And you can keep going, generating other potentials (like Helmholtz free energy) for other transformations.
