We have carried out electronic structure calculations on N interstitials in GaAs alloys using the first-principles plane-wave pseudopotential (PWPP) and projector augmented-wave (PAW) electronic structure methods in the framework of the density functional theory (DFT). Both the ultrasoft pseudopotential (USPP) method in connection with the generalized gradient approximation (GGA) and the PAW method in connection with the local density approximation (LDA) have been employed. Effects of the single nitrogen atom and nitrogen dimer related interstitial defects on the atomic and electronic structures of GaAs have been studied. Total energies, electronic band structures, and local densities of states have been evaluated. In general, energies of the defects with the NN dimer at the center of the Ga or As tetrahedron are more than 2 eV lower per nitrogen atom than those with a single N impurity at the same sites. We have also found that there are metastable defect candidates with a single N impurity in the middle of a particular edge of the Ga or As tetrahedra. Considering the modifications of the atomic structure of GaAs, our calculations show that the relaxations of the nearest neighbor atoms around the NN dimer at the center of the Ga or As tetrahedron, are essentially smaller than those around the single N impurity at the center of an edge of the Ga tetrahedron. Finally, these defect states induce drastic modifications into the electronic structure of GaAs. Interestingly, the NN dimer related defects cause noticeable changes only to the conduction bands near the conduction band edge, while the single N impurity related defects mainly modify the valence band edge and also induce localized and delocalized states into the band gap. Notably, the NN dimer and N impurity related defects lead to redshift and blueshift behavior of the band gap, respectively. The blueshift behavior is tentatively supported by the recent photoluminescence experiments.