What is meant by initial conditions, and zero initial conditions ... how do they differ?

Initial conditions are conditions at an initial time (t=0), from which a given set of mathematical equations or physical system evolves. The number of initial conditions that are required for a given differential equation will depend upon the order of the differential equation.

On the other hand, zero initial initial conditions are the initial conditions that are zero at time t=0. For instance the distance traveled by a runner at time t=0 is zero distance, x=0. It is where the input reference signal is zero, resulting to zero stored energy, no initial movements of parts, so the system is at rest.