A friend of mine asked me to explain science using a "top down" approach. Here's what I came up with:
Science is fundamentally a systematic way of determining "the rules." We posit the existence of underlying physical law (how the universe works) and science seeks to approximate that law with models. Models include theories, laws, hypotheses and facts. Commonly, theory and law refer to models that are accepted and largely validated (even if there are more accurate models). Facts are assertions that are modelled to be (believed to be) logically true. As you can tell, there's not a clear line differentiating theories, facts and laws in common usage. It doesn't really matter, though, because they're all models.
Science seeks to determine the models that best fit our observations of the world. The "truest" models, if you will. It does so through a process of refinement called the scientific method. A model, called a hypothesis, is tested through observation -- checking it's predictions against observations of the physical world. Generally, experiments are constructed to see how accurate a model is. If a model's accuracy cannot be determined through observation, the model is not "falsifiable" and is thus not scientific. That's not to say it's not in some sense true, but it means the model's accuracy can't be measured. (Note that traditionally people talk of disproving hypotheses, but scientific models are not right or wrong, they are more or less accurate).
The classic non-scientific statement is "God exists." Taking God to mean "an omniscient, omnipotent being," there's no way to measure God's existence. By being omnipotent and omniscient, God can exist but prevent any observation of evidence of God's existence. Since it's not falsifiable, the existence of God is not a scientific question.*
*I don't take this to mean that God doesn't exist, just that such existence is outside the purview of science. Some people do believe that non falsifiable statements cannot be true.
Often times, just observing the world is it happens isn't enough determine how accurate a model is. There are often too many complicating factors like weather that constantly change. So scientists set up experiments that attempt to control as many factors as they can. Then they vary the factors to see if they have any impact on the result.
It turns out that our best scientific models all have drawbacks. Quantum Mechanics doesn't do gravity right and is too computationally expensive to be applied on large scales. General Relativity works well on large scales but not small ones and it doesn't do gravity entirely right, either. Newton's laws work really well on human scales, but not so well on really small or large or fast moving scales. So we have a wide variety of models to describe a wide variety of things.
That's the root of science. 1) The universe has rules. 2) Models approximate those rules. 3) Models are tested and their accuracy measured through observation, particularly experiments. 4) Models are refined or new models created to be more accurate approximations of the underlying universal rules.
Math (and I'm including logic) is a formal way to describe relationships. As such, scientific models inevitably end up with mathematical descriptions.
Engineering shows up too. It's the application of the scientific method and scientific models to create new objects -- be they physical items like airplanes, information like software or social structures like governments. Without the scientific side, it's not engineering, it's artisanship. Another way to look at is that engineering is the application of scientific method and knowledge to art. So, naturally, engineers are both scientists and artists.*
*Yes, this is an idealization. But, really, without the scientist component an engineer is an artisan or an artist. And an engineer that doesn't create is more scientist or analyst. There is no shame in any of these occupations, though.