Differential equations are the type of equations that describe dynamic systems, whether they are naturally existing in the vast cosmos or humanly engineered in our universe. Such systems are characterized by having many variables that are changing with respect to others and therefore, the algebraic equations of school math are not the appropriate tool to use and the need for describing such variations arise. To describe variations, we recall using calculus for this, or more specifically differentiation, which is simply defined as the rate of change of a certain variable with respect to another; a simple example may be the rate of change of temperature with time in the event of heating a steel bar for instance. Of course, in large systems there are various variables and several changes (or derivatives) and the set of equations become exceptionally big one and in many cases, simplification is required to treat the system mathematically.
More on the DE page here (so far only the ordinary DE is added, partial DE to follow)