The structural stability, elastic properties and bonding behavior of Laves phase ZrMn₂ with C14, C15 and C36 structures as well as its hydride ZrMn₂-H₃ considering ferromagnetic (FM) ordering state were investigated by the first principles calculations. The calculated formation enthalpies and mechanical stability confirmed C14 structure is the stable crystal for ZrMn₂ in FM ground state. A better hydrogen storage behavior of C14 phase ZrMn₂ was predicted due to the obtained lower binding energy of hydrogen in ZrMn₂-H₃. In addition, there would be transformation of elastic behavior from ductility to brittleness, and increasing isotropy for ZrMn₂ when H atoms are absorbed into 12k sites of crystal structure to form hydride ZrMn₂-H₃. The calculated density of states, charge density distributions and Mulliken populations revealed that additional hybridizations and covalent interaction between Zr and H atoms would be introduced with the absorption of H atoms in C14-phase ZrMn₂.