Graphene, a one atomic thick planar sheet of carbon atoms, has a zero gap band structure with a linear dispersion relation. This unique property makes graphene a favorite for physicists and engineers, who are trying to understand the mechanism of charge transport in graphene and using it as channel material for field effect transistor (FET) beyond silicon. Therefore, an in-depth exploring of these electrical properties of graphene is urgent, which is the purpose of this dissertation. In this dissertation, the charge transport and quantum capacitance of graphene were studied. Firstly, the transport properties of back-gated graphene transistor covering by high dielectric medium were systematically studied. The gate efficiency increased by up to two orders of magnitude in the presence of a high top dielectric medium, but the mobility did not change significantly. The results strongly suggested that the previously reported top dielectric medium-induced charge transport properties of graphene FETs were possibly due to the increase of gate capacitance, rather than enhancement of carrier mobility. Secondly, a direct measurement of quantum capacitance of graphene was performed. The quantum capacitance displayed a non-zero minimum at the Dirac point and a linear increase on both sides of the minimum with relatively small slopes. The findings - which were not predicted by theory for ideal graphene - suggested that scattering from charged impurities also influences the quantum capacitance. The capacitances in aqueous solutions at different ionic concentrations were also measured, which strongly suggested that the longstanding puzzle about the interfacial capacitance in carbon-based electrodes had a quantum origin. Finally, the transport and quantum capacitance of epitaxial graphene were studied simultaneously, the quantum capacitance of epitaxial graphene was extracted, which was similar to that of exfoliated graphene near the Dirac Point, but exhibited a large sub-linear behavior at high carrier density. The self-consistent theory was found to provide a reasonable description of the transport data of the epitaxial graphene device, but a more complete theory was needed to explain both the transport and quantum capacitance data.