New gauges are introduced. The potentials, vector and scalar, in these gauges are obtained in closed forms by the Green's function method. These closed form solutions are explicity expressed only in terms of the charge and current densities. The physical interpretation is on how potentials propagate from the charge and current densities. The Coulomb gauge and the Lorentz gauge are special cases of a new gauge defined in this paper. It is called the complete α-Lorentz gauge. The scalar potential propagates at speed αc from the charge density for any positive α. When α is one, the usual solutions for the Lorentz gauge are recovered. When α is not one, our results show that, in order to satisfy the requirement that electromagnetic fields be gauge invariant and in order to conform to Maxwell's interpretation that electromagnetic fields propagate at speed c from the charge and current densities (we only consider the vacuum), the vector potential must contain two mathematically and physically independent gradient components. Furthermore, one such component must propagate at speed αc while the other must at speed c from charge and current densities. Our discussions on the Coulomb gauge are based on the results obtained by letting α go to (positive) infinity. Guided by Maxwell's interpretation, we introduce a new decomposition of the vector potential in the Lorentz gauge into a longitudinal and a transverse component. For an arbitrary charge and current distribution, it is shown that the transverse component will generate all the fields only in the radiation zone. However, for a point charged particle, the transverse component only generates the “free fields” everywhere in the instantaneous rest frame of the charged particle.