From the viewpoint of engineering applications, time delay is useful for constructing a chaotic signal generator, which is the major part of diverse potential applications. Although different mathematical models of time-delay systems have been known, few models can exhibit chaotic behaviors. Motivated by attractive features and potential applications of time-delay models, a new chaotic system with a single scalar time delay and a nonlinearity described by a saturation function is proposed in this chapter. Nonlinear behavior of the system is discovered through bifurcation diagrams and the maximum Lyapunov exponent with the variance of system parameters. Interestingly, the system shows double-scroll chaotic attractors for some suitable chosen system parameters. In order to confirm the correction and feasibility of the theoretical model, the system is also implemented with analog electronic circuit. Finally, a practical application of such circuit is discussed at the end of this chapter.