I am assuming that, by "level of measurement" you mean the "nominal, ordinal, interval, ratio" system introduced by Stevens and abused ever since.
First, I think you should be aware that these levels aren't exhaustive - that is, there are measurements that don't fit, see my blog post on this.
Notwithstanding the above, we can say some things:
Nominal with anything - Unless the nominal variable takes only 2 levels, correlation doesn't really make sense. There are measures of association, but not correlation.
Ordinal with ordinal, interval or ratio - It will often be best to use Spearman's rank correlation.
Interval or ratio with interval or ratio - Here, Pearson's correlation is a good place to start, but you need to be aware of other possible problems (look up Anscombe's quartet for some problems that can occur).