The A (1)Sigma(+) potential energy curve of AgH is studied by means of the second-order multistate multireference perturbation theory including the spin-orbit and relativistic effects. The anomalous behavior of the vibrational energy levels observed in experiment is reproduced well by theory. An analysis of the A (1)Sigma(+) wave function shows that at the internuclear distance of 4-6 Angstrom, a partial electron transfer from the 5p(z) orbital of silver to the 1s orbital of hydrogen occurs. This admixture of the ionic-like [core] 4d(10)1s(H)(2) configuration occurring in A (1)Sigma(+) due to two avoided crossings, namely X (1)Sigma(+) with A (1)Sigma(+) and A (1)Sigma(+) with C (1)Sigma(+), is found to be responsible for the unusual shape of the A (1)Sigma(+) potential energy curve: the effective potential is a superposition of a Morse-like covalent interaction between Ag and H, and the electrostatic ionic-like interaction between Ag+ and H-. We present spectroscopic parameters, vibrational levels, and rotational constants computed for a large number of vibrational levels and observe good agreement with available experimental data. The equilibrium distance agrees within 0.01 Angstrom and the vibrational frequency within 60 cm(-1) for the state-specific calculations. Larger relative discrepancy is observed for omega(e)x(e), about 30 cm(-1), however the non-Morse-like nature of the energy curve makes it impossible to describe the levels only with omega and omega(e)x(e), so that direct comparison is not well defined.