Plane-wave ultrasoft pseudopotential calculations are performed to study the structural stability and the electronic structure of N- or C-monodoped $TiO_2$. Firstly, various models of nonmetal doped $TiO_2$ are optimized to calculate the defect formation energy of different dopants. It is found that the Ti-rich condition facilitates N- or C-doping. The nonmetallic oxide of C and N in high valence facilitates doping under the Ti-rich condition. The element and the oxide of C and N in low valence facilitate doping under the O-rich condition. Secondly, the energy band structure of doped $TiO_2$ is analyzed. The results show that N-doping generates a semi-filled shallow impurity energy level near the top of the valence band. C-doping generates three deep impurity energy levels within the band gap which easily become the recombination centers of electrons and holes. Finally, the bonds between atoms are analyzed using the electron density map and the bond population. It is found that the electronegativity of impurities determines the position of the impurity energy level. These results help to understand the effects of nonmetal doping on the photocatalytic properties of $TiO_2$.