We are concerned with progressive waves propagating on a two-dimensional conducting fluid when a uniform electric field is applied in the direction perpendicular to the undisturbed free surface. The competing effects of gravity, surface tension, and electrically induced forces are investigated using both analytical and numerical techniques for an inviscid and incompressible fluid flowing irrotationally. We simplify the full Euler equations by expanding and truncating the Dirichlet-Neumann operators in the Hamiltonian formulation of the problem. The numerical results show that when the electric parameter is in a certain range, the bifurcation structure near the minimum of the phase speed is rich with Stokes, solitary, generalized solitary, and dark solitary waves. In addition to symmetric solutions, asymmetric solitary waves featuring a multi-packet structure are found to occur along a branch of asymmetric generalized solitary waves that itself bifurcates from Stokes waves of finite amplitude. The detailed bifurcation diagrams, together with typical wave profiles, are presented.