The 2nd Law of Thermodynamics states that any process that happens spontaneously will lead to an increase in the entropy of the universe. Entropy is given the symbol S.
Or
\[\Delta S_{\rm univ} > 0\]
By "spontaneously" we means that it is a process (a reaction for chemists) that tends to happen in the direction indicated (as written). Spontaneous reactions are poised to go forward provided there is nothing there to stop it. When a process or reaction is determined to be non-spontaneous, then it tends to not happen as written (because the reverse process is actually the spontaneous one). Chemical reactions that are spontaneous in a thermodynamic sense do not necessarily happen quickly or in many cases, happen at all. This stopping point that prevents spontaneous reactions from actually happening is a whole other area of chemistry called kinetics which deals with the rates of reactions. For a full understanding of chemical reactions you need to consider both the thermodynamics of the reaction and the kinetics of the reaction. But thermodynamics allows us to predict the range of possible things that could happen.
What do we mean by "universe"? When looking at changes in entropy, we have to consider not only changes for the system (the stuff we are studying), but also changes for the surroundings. We can add these two together to get the total entropy change for everything. In the largest sense, everything = the universe.
Next we need to think about the new state function, entropy.