I have read some articles about partial autocorrelation of time series and I have to admit, that I do not really comprehend the difference to a normal autocorrelation. It is often stated that the partial autocorrelation between $y_t$ and $y_t-k$ is the correclation between $y_t$ and $y_t-k$ with the influence of the variables between $y_t$ and $y_t-k$ removed? I do not understand this. If we calculate the correlation between $y_t$ and $y_t-k$ then anyways the variables in between are not consindered at all if you use the correlation coefficient for doing that. The correlation coefficient considers two variables only as far as I know.
This really confuses me. I hope you can help me on that. I'd appreciate every comment and would be thankful for your help.
Update: Can anyone try to explain how one could calculate autocorrelation and partial autocorrelation for a time series. I understood how to do this with a sample but not with a time series (because you need three variables according to the example here https://en.wikipedia.org/wiki/Partial_correlation). Do you know any example where this is done?