The goal of this paper is to introduce the quaternion orders. In Section ef{basic}, I give a brief introduction to quaternion algebra. Definitions and basic results necessary for the remaining sections are established in a very elementary way. Anyone who has learned one-year undergraduate algebra and knows the definition of tensor product should be able to read it without difficulties. In Section ef{general}, general properties about the quaternion orders are proven and the Eichler orders are introduced. In Section ef{rational}, special properties of the quaternion orders over the field of rational numbers are given. Section ef{local} and ef{global} are aimed to describe all the Eichler orders over number fields. In Section ef{local}, we give explicit formulations for the Eichler orders over the local field which is the completion of a number field with respect to one of its metric. In Section ef{global}, information about the local Eichler orders are collected to get an understanding about the Eichler orders over global fields. Some Eichler orders over the field of rational numbers are given explicitly.