Causation implies correlation but not necessarily the other way around. Correlation can be a result of confounding variables (X and Y are correlated because some latent variable Z is causing both X and Y).
I think you'd enjoy a read of Judea Pearl's "Causality". If you could get your hands on the book and read just the first chapter that should suffice.
I'll attempt a sweeping blurb but I can't guarantee its correctness. I'm probably going to screw up some statistical concepts. Also, I'm going to ignore sampling effects (which can cause spurious correlations and stuff).
Anyway, say you're observing a system. It has two variables: X and Y. You collect observational data and fit a statistical model to your data. Say we find a correlation (dependence) between X and Y in the data.