We have investigated the impact of the process parameters for the selective laser melting (SLM) of the stainless steel AISI 316L on its microstructure and mechanical properties. Properly selected SLM process parameters produce tailored material properties, by varying the laser’s power, scanning speed and beam diameter. We produced and systematically studied a matrix of samples with different porosities, microstructures, textures and mechanical properties. We identified a combination of process parameters that resulted in materials with tensile strengths up to 711 MPa, yield strengths up to 604 MPa and an elongation up to 31%, while the highest achieved hardness was 227 HV10. The correlation between the average single-cell diameter in the hierarchical structure and the laser’s input energy is systematically studied, discussed and explained. The same energy density with different SLM process parameters result in different material properties. The higher energy density of the SLM produces larger cellular structures and crystal grains. A different energy density produces different textures with only one predominant texture component, which was revealed by electron-backscatter diffraction. Furthermore, three possible explanations for the origin of the dislocations are proposed.