Laplace domain is the complex frequency domain which is denoted by s in your transfer function. Why is Laplace domain important? Because sometimes you will find that it is extremely efficient to analyze your system from zero/pole point of view, which can be extracted from the transfer function. In summary, state space model is a time domain analysis method, while laplace domain is a complex frequency domain analysis method. They are both very useful.
Oops - it looks like we had some font translation issues. Try looking at the slides later when they get posted in pdf format. All of the symbols above were supposed to be a "phase" symbol: