  Monolayer FeSe thin film grown on SrTiO_{3}(001) (STO) shows the sign of
T_{c}> 77 K, which is higher than the T_{c}-record of 56 K for the bulk
FeAs-based superconductors. However, little is known about the magnetic ground
state of FeSe, which should be closely related to its unusual
superconductivity. Previous studies presume the collinear stripe
antiferromagnetic (AFM) state as the ground state of FeSe, same to that in FeAs
superconductors. Here we find a novel magnetic order named "pair-checkboard
AFM" as the magnetic ground state of tetragonal FeSe. The novel pair-checkboard
order results from the interplay between the nearest, the next-nearest and the
unnegligible next-next-nearest neighbor magnetic exchange couplings of Fe
atoms. The monolayer FeSe in pair-checkbord order shows an unexpected
insulating behavior with a Dirac-cone-like band structure related to the
specific orbital order of d_{xz} and d_{yz} characters of Fe atoms, which could
explain recently observed insulator-superconductor transition. The present
results cast new insights on the magnetic ordering in FeSe monolayer and its
derived superconductors.
