This work discusses and presents the contributions of Galileo to hydrostatics, which were published in his book Discourse on buoyant bodies, in the frame of its historical context. Moreover, such contributions, still valid nowadays, are analyzed under the perspective of modern Fluid Mechanics. The application of theses principia is illustrated in the so-called Galileo Thermometer. Also, the role played by the body shape and surface tension in the buoyancy and sinking of solid material pieces more dense than the surrounding liquid is described in detail.