Elastic means "recoverable deformation" : The ability of a deformed material body to return to its original shape and size when the forces causing the deformation are removed.
linear elastic behaviour can be expressed as: Stress = (stiffness) * (strain)
Viscous means : The resistance to the time-rate-of-change of strain. It is the induced stresses due to the rate of strain.
linear viscous behaviour can be expressed as: Stress = (viscosity) * (rate of strain)