What's System Science and Why Should We Care?

It is very common for people to use the idea of a "system" pretty freely when discussing their ideas, projects, and problems .  Alas, they often have a pretty fuzzy idea about what a system is and how that perspective can be put to work.   "Systems Science" offers a concise definition of a system that is easy to contrast with traditional analytics.  In this post I'd like to start at the beginning and try to create a clear mental image of what systems science is and how it provides useful insights.