Recently, the novel two-dimensional semiconductor material β-TeO₂ has been successfully synthesized in experiments and has demonstrated excellent optoelectronic properties, attracting growing attention and research interest. Using first-principles calculations based on density functional theory, we systematically investigate the structural stability, electronic properties, and magnetism of the monolayer β-TeO₂ containing intrinsic point defects (vacancies, interstitials, and antisite defects) and substitutional defects. A number of physical quantities, such as the defect formation energy, electronic band structure, projected density of states, partial charge density, and magnetic moments, are calculated. It is found that, under oxygen-poor conditions, the most readily formed vacancy, interstitial, antisite, and substitutional defects are V_O1, O_i, TeO₂, and FO, respectively. Under oxygen-rich conditions, the corresponding defects are V_O1, O_i, TeO₂ and Sb_Te. The introduction of vacancy defects (V_O1, V_O2, and V_Te), interstitial defects (Te_i), and antisite defects (O_Te, Te_O1, and TeO₂) does not induce magnetism in monolayer β-TeO₂, and its semiconducting characteristics are still preserved. However, these defects introduce multiple spin-degenerate defect states within the bandgap, resulting in a notable decrease in the bandgap size. In contrast, the interstitial oxygen defects (O_i) do not induce magnetism and in-gap defect states in monolayer β-TeO₂, the system preserve its semiconducting character and an essentially unchanged bandgap. Conversely, all substitutional defects (F_O1, F_O2, N_O1, N_O2, I_Te, and Sb_Te) induce magnetism in monolayer β-TeO₂ and generate spin-polarized defect states within the bandgap, transforming the system into a magnetic semiconductor. The corresponding magnetic moments are 0.66μB, 0.59μB, 0.67μB, 0.74μB, 0.77μB and 0.75μB, respectively. Furthermore, we present a detailed analysis of the origin and mechanism of the defect states and magnetic moments. Our study advances the understanding of defect properties in two-dimensional β-TeO₂ and offers a theoretical foundation for its applications in electronic and spintronic devices.