Linear trend models-are the residuals supposed to be correlated over time?

I dont understand this part. Could some one explain this to me? If the residuals are correlated, isnt it supposed to be a poor model? Or is this the difference between linear regression and time series analysis?

Linear regression is an estimation technique, while time series analysis more so refers to the type of data you are analyzing. You might use a linear regression in a time series analysis, or you might use another technique (autoregressive model, exponential smoothing model, or one of many other techniques).

If you use a linear regression, one of the assumptions is that the errors are not autocorrelated. As you said, in this case, autocorrelated errors invalidate the linear regression model (assumption violated). In time series analysis, the residuals are more likely to be autocorrelated simply because the data are more prone to it. How this impacts your analysis will depend on the techniques you choose to analyze the data.

Hope this helps.