We examine thermal oscillation and resonance (with respect to time) described by the dual-phase-lagging heat-conduction equations analytically. Conditions and features of underdamped, critically damped and overdamped oscillations are obtained and compared with those described by the classical parabolic heat-conduction equation and the hyperbolic heat-conduction equation. Also derived is the condition for the thermal resonance. Both the underdamped oscillation and the critically damped oscillation cannot appear if the phase lag of the temperature gradient tau(T) is larger than that of the heat flux tau(q). The modes of underdamped thermal oscillation are limited to a region fixed by two relaxation distances defined by rootalphatau(T)(root(tau(q)/tau(T)) + root(tau(q)/tau(T))-1) and rootalphatau(T)(root(tau(q)/tau(T))-root(tau(q)/tau(T))-1) for the case of tau(T)>0, and by one relaxation distance 2rootalphaiota(q) for the case of iota(T)=0. Here alpha is the thermal diffusivity of the medium.