There are actually more than 4 thermodynamic potentials - their number depends on the number of degrees of freeedom of the system.
Say your internal energy is a function of the three extensive variables $S, V, N$. Then, you can swap out each of these for a corresponding intensive one via Legendre transformation. In principle, this gets you $2^3 = 8$ potentials, but one of them is just a constant as its differential is 0 due to the Gibbs-Duhem equation (which follows from internal energy being a homogeneous function).
Now, in order to get an intuitive understanding of a given potential, you hold its intensive parameters fixed and vary the extensive ones, of which there is at least one. The potential is the capacity to release energy under change of those latter variables (eg via expansion work if you vary $V$ or heat transfer if you vary $S$).
For example, Helmholtz free energy is defined by the transformation $S\to T$. If you hold temperature fixed,
energy can still be transferred by all other means except heat the change in Helmholtz free energy will be sensitive to all means of energy transfer except heat, so it's the systems capacity to perform (mechanical and non-mechanical) work (at fixed temperature).
Finally, as you alluded to ("gibbs free energy is a thermodynamic potential which indicates whether or not a chemical reaction will occur spontaneously at constant pressure and temperature"), you can use the potentials to analyze systems via the principle of minimum energy.