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We apply the density-functional theory to study various phases (including non-magnetic (NM), anti-ferromagnetic (AFM), and ferromagnetic (FM)) in monolayer magnetic chromium triiodide (CrI₃). We found that (1) the introduction of magnetism in monolayer CrI₃ gives rise to metal-to-semiconductor transition; (2) the electronic band topologies as well as the nature of direct and indirect band gaps in either AFM or FM phases exhibit delicate dependence on the magnetic ordering and spin-orbit coupling; and (3) the phonon modes involving Cr atoms are particularly sensitive to the magnetic ordering, highlighting distinct spin-lattice and spin-phonon coupling in this magnet. First-principles simulations of the Raman spectra demonstrate that both frequencies and intensities of the Raman peaks strongly depend on the magnetic ordering. The polarization dependent A_1g modes at 77 cm^-1 and 130 cm^-1 along with the E_g mode at about 50 cm^-1 in the FM phase may offer a useful fingerprint to characterize this material. Our results not only provide a detailed guiding map for experimental characterization of CrI₃, but also reveal how the evolution of magnetism can be tracked by its lattice dynamics and Raman response.