The motions of charged particles in electromagnetic fields composed of two or more laser beams show a variety of forms due to the adjustable properties of electromagnetic fields. In this paper, we consider the periodic laser standing wave field composed of two laser beams with opposite propagating directions. The movement of electrons in the standing wave field shows a periodic behavior, accompanied with the obvious radiation, especially when electrons are captured by the laser standing wave field. This phenomenon has aroused much interest of us. Under the existing experimental conditions, the free electron beam with low energy from an electron gun or the relativistic electron beam generated from laser acceleration can be easily obtained and injected into the periodic standing wave field. In this paper, using the single-electron model and the classical radiation theory of charged particles, we study the motion and radiation processes of low and high energy electrons in the polarized laser standing wave field. The results show that when the direction of incident electrons with low-speed is perpendicular to the direction of the laser standing wave electric field, the one-dimensional nearly periodic motion of electrons evolves into a two-dimensional folded movement by gradually increasing the light intensity of the laser standing wave field, and the strong terahertz radiation at micrometer wavelength is produced. High energy electrons generate the high-frequency radiation with the wavelength at several nanometers when the incident direction of high energy electrons is perpendicular or parallel to the direction of the laser standing wave electric field. In the case of low-energy electron, the motion of electron, frequency and intensity of radiation are affected by the laser intensity. In the case of incident high-energy electrons, the laser intensity affects the intensity of electronic radiation, and the initial electron energy influences radiation frequency. The bigger the incident electrons energy, the higher the frequency of radiation is. #br#We can obtain electron beams with different energies by laser acceleration, and they can be promising small radiation sources for terahertz and X-ray by using the electron beam radiation in a laser standing wave field. These studies also provide a basis for experimental researches and the applications of electron radiation in a laser standing wave field.