We investigate the quantal dynamics of the electronic and nuclear wave packet of H-2(+) in strong femtosecond pulses (greater than or equal to 10(14) W/cm(2)). A highly accurate method which employs a generalized cylindrical coordinate system is developed to solve the time-dependent Schrodinger equation for a realistic three-dimensional (3D) model Hamiltonian of H-2(+). The nuclear motion is restricted to the polarization direction z of the laser electric field E(t). Two electronic coordinates z and rho and the internuclear distance R are treated quantum mechanically without using the Born-Oppenheimer approximation. As the 3D packet pumped onto 1 sigma(u) moves toward larger internuclear distances, the response to an intense laser field switches from the adiabatic one to the diabatic one; i.e., electron density transfers from a well associated with a nucleus to the other well every half optical cycle, following which interwell electron transfer is suppressed. As a result, the electron density is asymmetrically distributed between the two wells. Correlations between the electronic and nuclear motions extracted from the dynamics starting from 1 sigma(u) can be clearly visualized on the time-dependent "effective'' 2D surface obtained by fixing rin the total potential. The 2D potential has an ascending and descending valley along z = +/-R/2 which change places with each other every half cycle. In the adiabatic regime, the packet starting from 1 sigma(u) stays in the ascending valley, which results in the slowdown of dissociative motion. In the diabatic regime, the dissociating packet localized in a valley gains almost no extra kinetic energy because it moves on the descending and ascending valleys alternately. Results of the 3D simulation are also analyzed by using the phase-adiabatic states \1] and \2] that are adiabatically connected with the two states 1 sigma(g) and 1 sigma(u) as E( t) changes. The states \1] and \2] are nearly localized in the descending and the ascending valley, respectively. In the intermediate regime, both \1] and \2] are populated because of nonadiabatic transitions. The interference between them can occur not only at adiabatic energy crossing points but also near a local maximum or minimum of E(t). The latter type of interference results in ultrafast interwell electron transfer within a half cycle. By projecting the wave packet onto \1] and \2], we obtain the populations of \1] and \2], P-1 and P-2, which undergo losses due to ionization. The two-state picture is validated by the fact that all the intermediates in other adiabatic states than \1] and \2] are eventually ionized. While E( t) is near a local maximum, P-2 decreases but P-1 is nearly constant. We prove from this type of reduction in P-2 that ionization occurs mainly from the upper state \2] (the ascending well). Ionization is enhanced irrespective of the dissociative motion, whenever P-2 is large and the barriers are low enough for the electron to tunnel from the ascending well. The effects of the packet's width and speed on ionization are discussed.