Super Simply, Correlation

Nobody explains correlation properly, and it's very easy to do.

A 0.5 correlation means that when something goes up, the thing it is 0.5 correlated with goes up half the time, and the other half of the time, it doesn't.
That's it and that's all.

Probably obvious: a 0 correlation means they're unrelated, and a 1 correlation means you're measuring the same thing twice. An identity relationship. Being a hatchling is has a 1.0 correlation with being a hatchling. A negative correlation, -1, just means you're measuring one of the things backwards. Growing out of being a hatchling has a -1 correlation with staying a hatchling; every time one goes up, the other goes down. In other words, functionally speaking correlation varies between 1 and 0, as the negative is nothing but a mirror of the positive.

So, example 0.5 correlation. You flip a coin, you get $1 every time it's tails, and $0 when it's heads. Getting more money has a 0.5 correlation with flipping the coin. 

How much it goes up is not relevant to this number. If you get $5 on a heads instead of $1 on a tails, then a flip is still 0.5 correlated with getting more money. It's about frequency, not magnitude. It's important to pay attention to the wording of the correlation; in this situation, there's a -0.5 correlation with your money going down. Same situation, same report, but a different wording makes it feel very different.

You should frequently use nonlinear correlation, but nobody does except physicists, who only deal with nonlinear correlations of 1 or -1 which they can express as simple equations.