Correlation: how to interpret confidence intervals in scatterplots? I have decided to represent the relationship between two variables with a scatter plot in R.  Now I have a question for you: how do I interpret the coloured area (the 95% confidence interval)? The confidence interval becomes somewhat narrower towards the middle and is somewhat wider at the "edges". What does that tell me?

Not my own scatterplot

You've correctly identified that the shaded region is likely a confidence interval (the confidence level is unknown but it is reasonable to assume it is 95%).

how do I interpret the coloured area

The interpretation of a confidence interval remains a hotly contested matter in many circles.  We can start with a definition.  A 95% confidence interval is an interval estimator which has the property that 95% of the time it is constructed (from the same data generating processes, under ideal conditions) it will contain the true estimand.  Suppose we are regression $y$ onto $x$ and we are interested in estimating the conditional mean of $y$ given $x=x_0$.  This would simply be $\hat{\beta}_0 + \hat{\beta}_1x_0$.  However, our estimate might change from sample to sample (because our sample changes).  If we repeated the experiment which generated the pairs $(x,y)$ which were used to fit the model ad infinitum and computed the confidence intervals, then on average 95% of them would contain the true conditional mean $\beta_0 + \beta_1 x$.
Now, one could understand if this is not a satisfactory explanation of how to interpret the interval.  One popular method I like for interpretation of a given interval (and one that appears here, in this very lengthy and quality discussion on the matter) is that the confidence interval provides us with values of the parameter (or in this case, the conditional mean) which are compatible with the observed data.
