The study of how to structure an information-gathering exercise where variation is present.
The design of experiments (DOE) is the study of how to structure an information-gathering exercise where the relationship between variables is not deterministic (i.e., at least some random variation is present). In statistics, controlled experiments are often implied, meaning that study units are randomly assigned to conditions in which the level of a variable[s] has been independently manipulated (minimally, and prototypically, to create a treatment condition and a control condition).
In the design of experiments, the experimenter is often interested in the effect of some process or intervention (the "treatment") on a response variable for some objects (the "experimental units"). True experiments are typically thought necessary for inferring causality, although there is a field of study within statistics that concerns itself with how causality may be nonetheless inferred from observational data.