A dynamical system exhibits chaos if it has solutions that appear to be quite random and the solutions exhibit sensitive dependence on initial conditions.
The sensitive dependence means that if you performed two experiments evolving the dynamical system where the only difference between the experiments was a slight difference in initial conditions, the resulting two trajectories would become very different as time progresses. (Since the initial conditions completely determine the future states, the experiments would be exactly the same if the initial conditions were identical.) In a real-world system, one wouldn't typically know initial conditions exactly, so chaos in a dynamical system severely limits the ability to predict the future of the system (think weather prediction).
The fact the solutions “appear quite random” means that if one were to start a whole bunch of experiments with different initial conditions that were all tighly clustered together, the resulting trajectories would eventually be spread all over the state space.