What is the definition of Computer Science, and what is the Science within Computer Science?
Computer science is a misnomer - there is actually no "science" in computer science, since computer science is not about observing nature. Rather, parts of computer science are engineering, and parts are mathematics.

The more theoretical parts of computer science are purely mathematical. For example, what is a good algorithm for sorting? How do we define the semantics of programming languages? How can we be sure that a cryptographic system is secure?

When computer science gets applied, it becomes more like engineering. For example, what is the best way to implement a matrix multiplication algorithm? How should we design a computer language to facilitate writing large programs? How can we design a cryptographic system to protect online banking?

In contrast, science is about laws of nature, and more generally about natural phenomena. The phenomena involved in computer science are man-made. Some aspects of computer science can be viewed as experimental in this sense, for example the empirical study of social networks, the empirical study of computer networks, the empirical study of viruses and their spread, and computer education (both teaching computer science and using computers to teach other subjects). Most of these examples are border-line computer science, and are more properly multidisciplinary. The closest one gets to the scientific method in computer science is perhaps the study of networks and other hardware devices, which is mainstream in the subarea known unofficially as "systems".

These examples notwithstanding, most of the core of computer science is not science at all. Computer science is just a name - it doesn't need to make sense.

As for the scope of computer science, the best definitions is perhaps: that which computer scientists do. Computer science, like every other academic discipline, is a wide area, and it is difficult to chart completely. If you want a sampling of what people consider computer science, you can look at the research areas of your faculty.