Doing correlation on one variable vs many I'm a programmer with little statistical background, and I'm trying to create something similar to what facebook did recently (with other data):
http://www.facebook.com/notes/facebook-data-team/whats-on-your-mind/477517358858
That is to be able to find correlation between one variable (age) and bunch of other variables or categories (type of words).
For the first chart, age vs. word type, I'm guessing they started out with data that looked like:
(header)
Status Update, Age, Word Cat1 percentage, Word Cat2 percentage, Word Cat3 percentage, etc..
and 1 million of such rows.
From what I understand correlation allows you to compare a variable to another variable, so how can I compare one variable (age) to a bunch of others?
 A: To help you get started with the visualization, here is a snippet of R code with simulated data (a matrix with age and counts for 20 words, arranged in columns, for 100 subjects). The computation are done as proposed my @mbq (correlation).
n <- 100  # No. subjects
k <- 20   # No. words
words <- paste("word", 1:k, sep="")
df <- data.frame(age=rnorm(n, mean=25, sd=5), 
                 replicate(k, sample(1:10, n, rep=T)))
colnames(df)[2:(k+1)] <- words
robs <- sort(cor(as.matrix(df))[-1,1])

library(lattice)
my.cols <- colorRampPalette(c("red","blue"))
res <- data.frame(robs=robs, x=seq(1,20), y=rep(1,20))

trellis.par.set(clip=list(panel="off"), axis.line=list(col="transparent"))
levelplot(robs~y*x, data=res, col.regions=my.cols, 
          colorkey=F, xlab="", ylab="", scales=list(draw=F),
          panel=function(...) {
            panel.levelplot(...)
            panel.text(x=rep(1, k), y=seq(1, k), lab=rownames(res))
         })


The above picture was saved as PDF, setting the margins to 1, and croped with pdfcrop from my TeXLive distribution.
pdf("1.pdf")
op <- par(mar=c(1,1,1,1))
(...)
par(op)
dev.off()

I guess it would not be too difficult to make a similar looking chart with barchart() from lattice, or ggfluctuation() or any other qplot() from ggplot2.
A: Correlation is a rather vague word meaning the fact that one variable is dependent on the other; in many cases this is just a synonym for Pearson correlation coefficient, which assumes linear dependence (i.e. $y=A \cdot x+B$), so things like "when x increses, y increases" (cor-> +1) or "when x increases, y decreases" (cor-> -1).
Regression on the other hand is a problem of finding a function that describes dependence between variables; in a linear problem it would be finding values of A and B, but in general it can be anything -- finding a period of some periodic phenomenon by fitting it with $y=\sin(\omega t+\phi)$ is also a regression.
Yet regression can be used to assess correlation -- if some regression model  fits the data well and in a significant way, it means there is correlation; this can be numerised with explained variance for instance. (In general it is a long and complex story)
That's all for oversized intro; going back to your question, this Facebook page shows Pearson correlations, which can be calculated without any regression with cor R function (also the formula is quite simple and available on the Wiki page). And as David wrote, counting correlation between $N$ variables boils down to counting correlations of all possible pairs.
