Last year's summary on correlational analysis is pretty thorough, so we'll largely use it. I just want to add a few things for this year:

First, here's a quote from the book Think of a Number (by Malcolm E. Lines, 1990) that puts the technique of partial correlation in some perspective:

It is probably the statistician's most difficult task of all to assure himself or herself that no unrecognized confounding factor lies hidden in the sampling groups which are being tested (p. 37).

Also, here are some correlation-related songs, for which I've written lyrics...

Fitting the Line (from one of my methods pages)

Restriction in the Range
Lyrics by Alan Reifman
(May be sung to the tune of “Laughter in the Rain,” Sedaka/Cody)

Why do you get such a small correlation,
With variables you think should be related?
Seems you’re not studying the full human spectrum,
Just looking at part of bivariate space,
All kinds of thoughts start to race, through your mind…

Ooh, there’s restriction in the range,
Dampening the slope of the best-fit line,
Ooh, I can correct r for this,
Put a better rho estimate in its place


Partial Correlation
Lyrics by Alan Reifman
(May be sung to the tune of “Crystal Blue Persuasion,” James/Vale/Gray)

Studying variables, with potential confounds,
Don’t want a paper, where confusion abounds,
There’s a technique, now, for consideration,
What you need to use, is partial correlation,

The connection of interest, is between A and B,
Each may be related, to the third factor, C,
There’s a formula out there, analysts would approve,
The influence of C, now, it will remove,

(Instrumental build-up)

Partial correlation,
For a strict-er determination,
Partial correlation,
A simple, calculation,

You’re purging C’s variance, from A and from B,
Thus from C’s linkage, the others are free,
The new A and B, now, you test for correlation,
The r, yes-sirree, AB-dot-C…

Partial correlation…

(Instrumental)

Partial correlation,
For a strict-er determination,
Partial correlation,
A simple, calculation,

Partial correlation ….
(Fade out)

[This website is excellent on partial correlation.]

Correlation (about correlation and causality, from one of my methods pages)