What is expressed by the terms zeroth-, first-, second-, third-, etc. order of correlation? Thanks!
Here is a nice resource for understanding these issues. It's excellent; you should read it thoroughly.
However, I will give a quick introduction. Imagine you have 3 variables, $x$, $y$ and $z$. You are primarily interested in the relationship between $x$ and $y$, but you know that $y$ is also related to $z$, and that unfortunately, $z$ is confounded with $x$. If you simply wanted to know the strength of the relationship, Pearson's product-moment correlation coefficient $r$ is a useful effect size measure.
In this situation, you could simply ignore $z$ and compute the correlation between $x$ and $y$ (this is not really a good idea, as the value would be biased). Since you have controlled for nothing, this is a 'zero-order' correlation.
You might opt instead for a more conscientious approach and control for the confounding with $z$, by partialling out $z$. (One conceptually clear way to do this, albeit not computationally optimal, is to regress $y$ onto $z$, and $x$ onto $z$, and then compute the correlation between the residuals of the two models.) Because you have controlled for one variable, this would be a 'first-order' partial correlation.
I have never seen such a thing in practice, but if you partialled out 17 other vairables, you would have a 'seventeenth-order' partial correlation. The linked website is very informative, with examples, multiple formulas and diagrams; go read it.