A lumped model is just a mathematical model with a finite number of degrees of freedom (DOF). That could be a finite difference grid or a finite element model (possibly with millions of DOF) or something like a simple "rigid masses connected by massless springs" dynamics model with a small number of DOF (even as small as just 1 DOF).
A continuous model has an infinite number of DOF. The commonest type of continuous models are analytical (not numerical) solutions of ordinary or partial differential equations.