Proteins in solution form a number of condensed phases. Even omitting the amyloid structures formed after partial protein unfolding, these phases include crystals, polymers, and other solid aggregates, as well as dense liquids and gels. Some of these condensed phases underlie pathological conditions, while others play a crucial role in the biological function of the respective protein or are an essential part of its laboratory or industrial processing. In this review, we summarize the fundamentals and recent findings on the kinetics of nucleation of dense liquid droplets and crystals. We define the transition from nucleation to spinodal decomposition for these two phase transitions. We review the two-step mechanism of protein crystal nucleation, in which mesoscopic metastable protein clusters serve as precursors to the ordered crystal nuclei. The concepts and mechanisms reviewed here provide powerful tools for control of the nucleation process, by varying the solution thermodynamic parameters.