the difference between active and inactive ingredients

Source: Food and Drug Administration (FDA)

Active ingredients are those which are intended to have a therapeutic and pharmacologic activity.
An active ingredient is the main portion of a drug product intended to produce a therapeutic effect.
An active ingredient is any component that provides pharmacological activity or other direct effect in the diagnosis, cure, mitigation, treatment, or prevention of disease, or to affect the structure or any function of the body of humans or animals.

Inactive ingredients are added for the purpose of formulation compatibility and proper drug delivery (i.e., binding ingredients, disintegrants, etc...)
Inactive ingredients can also include flavors, color dyes, and are intended not to offer any pharmacologic activity.