This article presents, within the framework of general relativity, the principal features of spherically symmetric self-gravitating objects, either static (stars in equilibrium) or dynamic (stars in gravitational collapse). The Einstein equations are written in the so-called RGPS coordinates which are employed in some numerical studies of gravitational collapse. These coordinates generalize the Schwarzschild coordinates to the non-vacuum and non-static case. In terms of the 3+1 formalism of general relativity, they are defined by the maximal slicing and the radial gauge. They are used here in conjunction with particular hydrodynamical variables, as the fluid energy density and velocity both as measured by an observer who generalizes the classical Eulerian observer. The resulting equations for self-gravitating fluids have then simple expressions, of Newtonian aspect. For instance, the generalization of the Euler equation has a very simple form, each term of which can be easily interpreted. The discussion of the equations reveals a global quantity as the total mass-energy of the star. This quantity is conserved during the evolution, in agreement with the fact that there is no gravitational wave emission in spherical symmetry. Besides, the study of the characteristic curves of the system of hydrodynamical equations leads to a simple expression for the relativistic sound velocity. In the case of a vanishing velocity, the equations are reduced to the usual Tolman-Oppenheimer-Volkoff system, which governs the hydrostatic equilibriums in general relativity. Properties of these latter ones, as well as results about their stability, are reminded. An illustration is provided with the specific example of polytropes. The comparison with the Newtonian polytropes displays the major relativistic effects, as the existence of a maximum mass, whatever the stiffness of the equation of state may be.,This allows one to interpret the various maximum masses which appeared in the literature in terms of equation of state and theory of gravitation. For instance, the Chandrasekhar maximum mass for white dwarfs is entirely due to the softness of the equation of state, whereas the Oppenheimer-Volkoff maximum mass for neutron stars is a truly general relativistic effect. The dynamical case is illustrated by the analytical Oppenheimer-Snyder solution for the collapse into black hole of a homogeneous ball of pressureless matter. This solution is re-expressed here in terms of the RGPS coordinates instead of the classical Robertson-Walker ones, in order to analyse the behavior of the introduced metric and hydrodynamical variables, in a situation close to the realistic gravitational collapse of a star into black hole. The RGPS coordinates are able to follow the collapse up to the ''frozen star'' state, where no more subsequent evolution would be perceived by a remote observer. This leads to the conclusion that RGPS coordinates are valuable coordinates for numerical study of gravitational collapse in spherical symmetry.
