Thermodynamics is the study of energy, heat, and work transitions within chemical systems. It allows us to predict the spontaneity of reactions and the maximum work obtainable from a process.
The total energy stored within a system. According to the First Law, $\Delta U = q + w$, where $q$ is heat exchanged and $w$ is work done on the system.
The criteria for spontaneity at constant temperature and pressure. If $\Delta G < 0$, the reaction is spontaneous (Exergonic).
$\Delta U$ represents energy change at constant volume, while $\Delta H$ (Enthalpy) represents energy change at constant pressure, including the work done to displace the environment.
The index $x$ defines the process type: $x=1$ is Isothermal, $x=\gamma$ is Adiabatic, $x=0$ is Isobaric, and $x=\infty$ is Isochoric.
It is the "available" energy to do non-expansion work. It acts as the ultimate decider for whether a chemical reaction will proceed without external help.