We can break down entropy change as resulting from changes in specific properties of our system. There are changes in five things that will lead to a change in the entropy of the system.
Changes in temperature will lead to changes in entropy. The higher the temperature the more thermal energy the system has; the more thermal energy the system has, the more ways there are to distribute that energy; the more ways there are to distribute that energy, the higher the entropy. Increasing the temperature will increase the entropy.
Changes in volume will lead to changes in entropy. The larger the volume the more ways there are to distribute the molecules in that volume; the more ways there are to distribute the molecules (energy), the higher the entropy. An increase in volume will increase the entropy. This is typically only important for systems involving gases since they are the only materials that undergo large volume changes.
Changes in phase will lead to changes in entropy. Some phases have larger numbers of microstates and thus higher energy. Solids have the fewest microstates and thus the lowest entropy. Liquids have more microstates (since the molecules can translate) and thus have a higher entropy. When a substance is a gas it has many more microstates and thus have the highest entropy.
Mixing of substances will increase the entropy. This is because there are many, many more microstates for the mixed system than for the un-mixed system. More microstates means greater entropy. This is one of the examples from which the misconception that entropy is "disorder" arises.
Lastly, the entropy can change as the result of chemistry. Different molecules have different entropies. Thus it can be difficult to look at a reaction and guess if the entropy is going up or going down. However, in general if the products have a larger number of molecules than the reactants then the entropy is likely to increase. Additionally, if products are in phases of higher entropy than the reactants than the reaction is likely to have a higher entropy. For example
\[\rm 2Fe(s) + 3O_2(g) \rightarrow 2Fe_2O_3(s)\] \[ \Delta S_r < 0\]
The change in entropy for this reaction will be negative. This because the reactants have both a solid and a gaseous species while the product is simply a solid compound.
For these particular changes in entropy we will be doing quantitative calculations for the entropy changes that arise from temperature change, phase change, and chemistry.