In this work, we employed first-principles calculations to investigate the stability, electronic structures, mechanical and thermodynamic properties of Fe_xAl_yC_z compounds which were first observed in Fe-Cr-B-Al-C alloy. The results reveal that the Fe_xAl_yC_z compounds are thermodynamically stable, and the addition of C element improves the stability of Fe_xAl_yC_z compounds. These Fe_xAl_yC_z compounds display disparate anisotropy according to different calculated 3D curved surfaces of Young's modulus and anisotropic index. The Fe₃AlC compound has the biggest shear modulus, bulk modulus and Young's modulus with the values of 141.2 GPa, 212.6 GPa and 346.8 GPa, respectively, which indicates that the Fe₃AlC compound has the largest rigidity, lateral deformation resistance and good solid compressibility under static pressure. The ratio of G/B shows that FeAl and Fe₃AlC are brittle phases, while the Fe₃Al and FeAl₃ phases appear to be ductile. The calculated electronic structure reveals that the bonding characteristics of the Fe_xAl_yC_z compounds are the mixture of both metallic and covalent bonds, which also exhibits anti-bond effects. The Fe₃AlC compound has the biggest Debye temperature of 702.2 K. Meanwhile, the hardness of Fe₃AlC is 19.4 GPa which is three times of the Fe₃Al compound. The results obtained in this study provide a theoretical basis for the applications of Fe₃AlC compound in Fe-Cr-B-Al-C alloys.