Can you use normal correlation for vectors with only 2 (or 3), ordered, levels? When one wants to compute the correlation of two vectors of a continues variables, one uses pearson (or spearman) correlation.
But what should (can) one use for the case of two vectors with 2 (or 3) ordered levels only?  Is spearman enough, or does it require another method?
I remember coming across someone who once claimed to me that OR (odds ratio) is more fitting for such situations (for 2 by 2 tables, where order has no meaning), is this true?
Here is an example R code, for allowing of answers relating to the same example:
set.seed(10)

x2 <- sample(c(-1,1),50, T)
x3 <- sample(c(-1:1),50, T)
y3 <- sample(c(-1:1),50, T)
y2 <- sample(c(-1,1),50, T)

cor(x3,y3, method = c("spearman"))
cor(x2,y2, method = c("spearman"))
cor(x3,y2, method = c("spearman"))

p.s: for the 2 by 2 case, I followed from the comments that categorical "measures of association" is the term to look for.  However, Part of the time I am comparing 2 on 3 tables, where on the factor with 3 levels there is order - so I would like to take use of that information.
 A: A few thoughts: 


*

*There are many different binary-binary and ordinal-ordinal measures of association.
SPSS provides names and algorithms for many of them under proximities
and crosstabs.

*I'm also intrigued by  tetrachoric (binary-binary) and polychoric (ordinal-ordinal) correlations that
aim to estimate the correlation between theorised latent continuous variables.

*You can use Pearson's correlation. However, it is not always the most meaningful metric of association. Also, confidence intervals and p-values that assume continuous normal variables wont be perfectly accurate.

A: If you have more than two levels, you can use (M)CA:
http://en.wikipedia.org/wiki/Correspondence_analysis
A: The OR is a good measure of association, but sometimes people prefer a correlation coefficient for interpretation because it has a [-1, 1] scale.  
For binary variables, the Phi statistic provides Pearson's correlation (see Jeromy's comment).  Cramer's V is applicable when you have more than 2x2 cases.  For details, see the following references:


*

*Effect size

*Correlation

*Cramer's V
I've never used any of these, so hopefully someone will jump in and say if there are good reasons for preferring them.
