In the paper, a fractional-order Lozi map with discrete fractional calculus is analyzed. Firstly, dynamics of the map with different values of a parameter or a derivative order is studied. Hopf and Period-doubling bifurcations are observed. The map has the chaotic attractors with different topological structures. Secondly, when a system parameter and a derivative order are varied, bifurcations for the system are investigated and presented in a three-dimensional space. The dynamics behavior of the system changes from chaos to regular with an decrease of derivative orders, and changes from regular to chaos with an increase of derivative orders. Finally, the synchronization of the fractional-order discrete Lozi map is studied. The validity of the designed controllers is demonstrated by the numerical simulations.