Quite simply any system who's future state cannot be predicted from observation of its past and present states.
Weather systems are not examples of random systems they are examples of chaotic systems are defined as 'certain nonlinear dynamical systems that may exhibit dynamics that are highly sensitive to initial conditions (popularly referred to as the butterfly effect). As a result of this sensitivity, which manifests itself as an exponential growth of perturbations in the initial conditions, the behavior of chaotic systems appears to be random. This happens even though these systems are deterministic, meaning that their future dynamics are fully defined by their initial conditions, with no random elements involved. This behavior is known as deterministic chaos, or simply chaos.'
The main difference between the two is that a chaotic system's future state can be predicted bu looking at its past and present state and a random system's cannot.