What you understand by "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.