I've taken at least 3 different statistics courses, and I feel like I'm only now internalizing the core concepts. The course I took a month ago was key, because the prof started from the very beginning, and was very, very thorough. He explained a few things that I'm sure I learned many times before, but are often glazed over (I guess because they're deemed too simple to dwell on). Yet if these key ideas fly out of your head for even a moment, the whole enterprise starts to seem spooky and incomprehensible. One of these ideas is what I'm going to call randomosity. How random is randomosity? Well, not at all.
Statistics is simply math applied to the real world. But the real world is incredibly complicated, so complicated that it often seems like chaos - completely random. But what seems random isn't always really random! The key is discerning the random from the non-random.
This is possible because pure chance is possible to calculate with absolute accuracy (a funny thought eh?). On the other hand, natural phenomena with non-random causes and effects have so many variables and are so complex that even our most powerful models are only approximations. Chance, on the other hand, is known. The chance of flipping heads is always the same - exactly 50%. With a die you have exactly 1 in 6 chances of rolling a given number. Not sort of. Exactly. (Sorry lotto players.)
We're comparing what would happen under random circumstances to what we have actually observed, and then seeing to what extent they differ. This is the guiding principle of many statistical tests.
This site is pretty easy to understand: http://www.socialresearchmethods.net/