The Gibbs free energy is defined (in difference form) as $$\Delta G\equiv\Delta H-T\Delta S=\Delta U+P\Delta V-T\Delta S,$$
where $H$ is the enthalpy, $T$ is the temperature, $S$ is the entropy, $U$ is the internal energy, $P$ is pressure, and $V$ is volume. Note that two terms have been added to the internal energy. The first, $P\Delta V$, represents the work done on a surrounding constant-pressure atmosphere; it acknowledges the fact that our system takes up space and needs to do work on its surroundings simply to exist at a finite volume. This is useful because we often consider as systems the materials and processes around us, which are surrounded by air at 1 atm.
The second term, $T\Delta S$, represents energy provided by the surrounding constant-temperature environment; it acknowledges the fact that at nonzero temperature, there's always a nonzero likelihood that some microscale process can occur (such as an atom evaporating from the surface of condensed matter). You can think of the surroundings as providing energy for free, even fluctuatingly large energies, simply by acting as a large temperature reservoir. This concept is also useful because again, the materials and processes around us are at room temperature, perhaps, or some other well-regulated temperature. As a result, we often observe phenomena that require energy and therefore would be nonsensical if we operated under the equilibrium assumption of $\Delta U=0$.
The additional $T\Delta S$ implies that Nature—at least constant-temperature Nature—favors processes that release energy (such as strong bonding, which puts molecules in low-enthalpy states) but also favors processes that increase entropy (such as a formerly fixed atom that can now bounce around in the gas phase with an endless possibility of locations, speeds, and orientations). Which factor dominates depends on the coefficient of $\Delta S$, namely, the temperature $T$. At higher temperature, the equilibrium phase is always the higher-entropy phase, for instance. Oxidation reactions may reverse because of the benefit of creating a gas product. Condensed matter sublimates and evaporates and is cooled in the processes. None of this can be understood without considering the $T\Delta S$ term and by modifying our criterion of equilibrium (at constant pressure and temperature) from $\Delta U=0$ to $\Delta G=0$.