Entropy ($S$) is a measure of the disorder in a system.
In a closed system, entropy always increases over time. In an open system, energy can be added to a system to cause a decrease in entropy, but this is not necessarily a spontaneous reaction.
A reaction will be spontaneous if the change in free energy, $\Delta G$, is negative.
if entropy increases, and the reaction enthalpy is exothermic $(\Delta H<0)$ or weakly endothermic ($\Delta H>0$ and $\Delta H < T\Delta S$), the reaction becomes spontaneous according to the free energy equation.
Since the observable universe is essentially a closed system, entropy will continue to increase until the universe essentially ceases to exist.
This way an increase or decrease(sometimes) in the randomness can cause a reaction to occur.
Here is a table relating to spontaneity and signs of $\Delta G$: