The studies of spin phenomena in semiconductor low-dimensional systems have grown into the rapidly developing area of the condensed matter physics: spintronics. The most urgent problems in this area, both fundamental and applied, are the creation of charge carrier spin polarization and its detection, as well as electron spin control by nonmagnetic methods. Here, we present a review of recent achievements in the studies of spin dynamics of electrons, holes, and their complexes in the pump-probe method. The microscopic mechanisms of spin orientation of charge carriers and their complexes by short circularly polarized optical pulses and the formation processes of the spin signals of Faraday and Kerr rotation of the probe pulse polarization plane as well as induced ellipticity are discussed. A special attention is paid to the comparison of theoretical concepts with experimental data obtained on the n-type quantum well and quantum dot array samples.