I have a dataframe with many observations and many variables. Some of them are categorical (unordered) and the others are numerical.

I'm looking for associations between these variables. I've been able to compute correlation for numerical variables (Spearman's correlation) but :


I don't know how to measure correlation between unordered categorical variables.
I don't know how to measure correlation between unordered categorical variables and numerical variables.


Does anyone know how this could be done? If so, are there R functions implementing these methods?

It depends on what sense of a correlation you want.  When you run the prototypical Pearson's product moment correlation, you get a measure of the strength of association and you get a test of the significance of that association.  More typically however, the significance test and the measure of effect size differ.  

Significance tests:  


Continuous vs. Nominal:  run an ANOVA.  In R, you can use ?aov.  
Nominal vs. Nominal: run a chi-squared test.  In R, you use ?chisq.test.  


Effect size (strength of association):  


Continuous vs. Nominal:  calculate the intraclass correlation.  In R, you can use ?ICC in the psych package; there is also an ICC package.  
Nominal vs. Nominal: calculate Cramer's V.  In R, you can use ?assocstats in the vcd package.  

