When studying Kinetic chemistry (and most of the Physical chemistry topics), we often find definitions based upon the mathematical formalism. For example, Order of reaction is often defined as the sum of the partial order of a Rate law. But is there a way to define it without using the mathematical formalism? I mean, if we think microscopically, how would we define Order of reaction?
Historically, the concept of reaction order came from Ostwald's famous Lehrbuch of 1890s. An abridged translation is available in Internet Archive as "Outlines of General Chemistry". In this book, surprisingly, when a single reactant undergoes a transformation he does not call it a first order reaction. He designated rate of reactions controlled by two species as reaction of the second order.
Reaction of the Second Order. We come to a second kind of chemical process, regulated by a new law, when in its course two substances undergo change of quantity. We must again assume that the action is proportional to the concentration of each present.
Reactions of Higher Order. We must now consider reactions in which three or more substances take part.
So roughly, you can think of reaction order as the number of species in a chemical reaction which control the rate of reaction.
Coming to modern times, you have fractional reaction orders, and zero order and so on. For understanding you will need to see the concept of elementary and overall reaction orders. A reaction may be occurring in several steps. In such cases, you might end up in fractional exponents. However, once you are able to know the steps of that reaction, you will see that those elementary reactions don't have fractional exponents. You can then interpret, each elementary reaction in the original Ostwald's way.