Does the autocorrelation function have any meaning with a non-stationary time series?
The time series is generally assumed to be stationary before autocorrelation is used for Box and Jenkins modeling purposes.
@whuber gave a nice answer. I would just add, that you can simulate this very easily in R:
Which ends up looking somewhat like this:
So you can easily see that the ACF function trails off slowly to zero in the case of a non-stationary series. The rate of decline is some measure of the trend, as @whuber mentioned, although this isn't the best tool to use for that kind of analysis.
In its alternative form as a variogram, the rate at which the function grows with large lags is roughly the square of the average trend. This can sometimes be a useful way to decide whether you have adequately removed any trends.
(You can think of the variogram as the squared correlation multiplied by an appropriate variance and flipped upside down.)