State-space system representation lays the foundations for modern control theory. It solves many of the limitations of the classical control theory in which transfer functions were used to asses the behavior of a closed loop system.
A state-space model describes the behavior of a dynamic system as a set of first order ordinary differential equations (ODE).
If a dynamic model is described by a higher order ODE, using state-space, the same model can be described as a set of coupled first order ODEs.
The internal variables of the state-space model are called state variables and they fully describe the dynamic system and its response for certain inputs.
The numbers of state variables of the state-space model is equal with the highest order of the ODE describing the dynamic system.
State variables can also be defined as the smallest set of independent variables that completely describe the system.