The physical phenomena of plasma plume generation and plasma expansion by target-laser interaction are considered for moderate laser power density. The kinetics of target vaporization, atom ionization, and plasma heating are described. The mechanism of electric sheath formation near the surface, electron emission from the target, and the electrical breakdown phenomena by laser irradiation are analyzed. The plasma expansion is described taking into account the near target plasma structure and the absorption of the laser radiation. The mechanisms accelerating the plasma and generating an electric field in it are discussed. The work reviews experiments and theoretical models and summarizes the results in order to understand the measurements and to discuss open questions. As example, the plasma parameters (electron temperature and density, degree of ionization, and plasma velocity) are calculated for an Ag target using the developed model that considers self-consistently the target heating, kinetics of target evaporation, plasma heating, and ion flux to the target. The calculated ion velocity in the expanding plasma jet is in accordance with the measurement. The ion energy linearly depends on the ion charge state, as observed experimentally.