A theoretical and experimental study of the electron structure and nature of the chemical bonds in FeTe compounds in antiferromagnetic (AFM) and paramagnetic phases. It is established that the nature of the chemical bonds is mainly metallic, and the presence of covalent bonds Fe-Te and Te-Te helps to stabilize the structural distortions of the tetragonal phase of FeTe in the low-temperature region. It is found that the bicollinear AFM structure corresponds to the ground state of the FeTe compound and the calculated value of the magnetic moment MFe = -2.4μB is in good agreement with the data from neutron diffraction measurements. At the same time, the Fermi surface (FS) of the low-temperature AFM phase is radically different from the FS of the paramagnetic FeTe. Reconstructing the FS can lead to a sign change of the Hall coefficient observed in FeTe. The calculation results serve as evidence of the fact that the electron structures and magnetic properties of FeTe are well-described by the model of itinerant d-electrons and the density functional theory (DFT-GGA).