The electrochemical N2 reduction reaction (NRR) under ambient conditions is attractive in replacing the current Haber-Bosch process toward sustainable ammonia production. Metal-heteroatom-doped carbon-rich materials have emerged as the most promising NRR electrocatalysts. However, simultaneously boosting their NRR activity and selectivity remains a grand challenge, while the principle for precisely tailoring the active sites has been elusive. Herein, we report the first case of crystalline two-dimensional conjugated covalent organic frameworks (2D c-COFs) incorporated with M-N4-C centers as novel, defined, and effective catalysts, achieving simultaneously enhanced activity and selectivity of electrocatalytic NRR to ammonia. Such 2D c-COFs are synthesized based on metal-phthalocyanine (M = Fe, Co, Ni, Mn, Zn, and Cu) and pyrene units bonded by pyrazine linkages. Significantly, the 2D c-COFs with Fe-N4-C center exhibit higher ammonia yield rate (33.6 μg h-1 mgcat-1) and Faradaic efficiency (FE, 31.9%) at -0.1 V vs reversible hydrogen electrode than those with other M-N4-C centers, making them among the best NRR electrocatalysts (yield rate >30 μg h-1 mgcat-1 and FE > 30%). In situ X-ray absorption spectroscopy, Raman spectroelectrochemistry, and theoretical calculations unveil that Fe-N4-C centers act as catalytic sites. They show a unique electronic structure with localized electronic states at Fermi level, allowing for stronger interaction with N2 and thus faster N2 activation and NRR kinetics than other M-N4-C centers. Our work opens the possibility of developing metal-nitrogen-doped carbon-rich 2D c-COFs as superior NRR electrocatalyst and provides an atomic understanding of the NRR process on M-Nx-C based electrocatalysts for designing high-performance NRR catalysts.