We study the action of the elements of the mapping class group of a surface of finite type on the Teichmüller space of that surface equipped with Thurston's asymmetric metric. We classify such actions as elliptic, parabolic, hyperbolic and pseudo-hyperbolic, depending on whether the translation distance of such an element is zero or positive and whether the value of this translation distance is attained or not, and we relate these four types to Thurston's classification of mapping classes. The study is parallel to the one made by Bers in the setting of Teichmüller space equipped with Teichmüller's metric, and to the one made by Daskalopoulos and Wentworth in the setting of Teichmüller space equipped with the Weil-Petersson metric.