For single input , single output linear systems we define 'type' according to the number of integrators (number of free 's' in the denominator) in the open loop transfer function. A type zero system would have zero integrators, a type one, one integrator, and so forth.
The significance of classifying system types gives you an idea of how the system will behave to a specific type of input signal if you apply simple, proportional feedback to the open loop system.
For a type 0 system with step input you would see a finite steady state error, proportional to the loop gain.
For a type 1 system with step input you would see a zero steady state error.
This page provides a more complete summary (table) for type systems (denoted by N) and the expected output.
For nonlinear systems you cannot generally apply the same principles so a 'type' is not defined.
For type you might be confusing the 'type' of control compensator (filter): P, I, PI or PID types which more often refer to the control compensator that's applied in the control loop to control a system. As an open loop transfer function yo determine type (0, 1,2 , etc.) you can consider the compensator together with the plant as the open loop transfer function - to define the type you would get after closing the loop.