Spontaneous process
A spontaneous process is a chemical reaction in which a system releases free energy (most often as heat) and moves to a lower, more thermodynamically stable, energy state. The sign convention of changes in free energy follows the general convention for thermodynamic measurements, in which a release of free energy from the system corresponds to a negative change in free energy, but a positive change for the surroundings.
A process that is capable of proceeding in a given direction, as written or described, without needing to be driven by an outside source of energy. The term is used to refer to macro processes in which entropy increases; such as a smell diffusing in a room, ice melting in lukewarm water, salt dissolving in water, and iron rusting.
The laws of thermodynamics govern the direction of a spontaneous process, ensuring that if a sufficiently large number of individual interactions (like atoms colliding) are involved then the direction will always be in the direction of increased entropy (since entropy increase is a statistical phenomenon).

# Overview
For a reaction at constant temperature and pressure, the Gibbs free energy is:
a negative G would depend on the sign of the changes in enthalpy (H), entropy (S), and the magnitude of the absolute temperature (in kelvins).  Changes in the sign of G cannot be changed directly by temperature, because it can never be less than zero.
When S is positive and H is negative, a process is spontaneous
When S is positive and H is positive, a process is spontaneous at high temperatures, where exothermicity plays a small role in the balance.
When S is negative and H is negative, a process is spontaneous at low temperatures, where exothermicity is important.
When S is negative and H is positive, a process is not spontaneous at any temperature, but the reverse process is spontaneous.
The second law of thermodynamics states that for any spontaneous process the overall change in entropy of the system must be greater than or equal to zero, yet a spontaneous chemical reaction can result in a negative change in entropy.  This does not contradict the second law however, since such a reaction must have a sufficiently large negative change in enthalpy (heat energy) that the increase in temperature of the reaction surroundings (considered to be part of the system in thermodynamic terms) results in a sufficiently large increase in entropy that overall the change in entropy is positive. That is, the S of the surroundings increases enough because of the exothermicity of the reaction that it overcompensates for the negative S of the system, and since the overall S = Ssurroundings + Ssystem, the overall change in entropy is still positive.  
Another way to view the fact that some spontaneous chemical reactions can lead to products with lower entropy is to realize that the second law states that entropy of a closed system must increase (or remain constant). Since a positive enthalpy means that energy is being released to the surroundings, then the 'closed' system includes the chemical reaction plus its surroundings. This means that the heat release of the chemical reaction sufficiently increases the entropy of the surroundings such that the overall entropy of the closed system increases in accordance with the second law of thermodynamics.
In simplified terms, a spontaneous reaction is one in which a meaningful amount of products are formed from the reactants without the addition of a catalyst.  However, a "nonspontaneous" reaction may still proceed, but will not convert the reactants into an appreciable amount of products. The reaction may proceed to its equilibrium point, but its equilibrium is very small (possibly on the order of 10-23 or smaller).  For example, table salt (NaCl) will not spontaneously separate into individual ions (Na+ and Cl-), unless it is greatly heated or forcibly separated by electrolysis, yet small, immeasurable amounts of the ions may form.  Additionally, just because a chemist may call a reaction spontaneous does not mean the reaction happens with great speed. For example, the decay of diamonds into graphite is a spontaneous process but this decay is extremely slow and takes millions of years. Thus the rate of a reaction is independent of its spontaneity, and instead depends on the chemical kinetics of the reaction.