In my experience users of exponential smoothing have often limited transparency in how the various smoothing parameters interact. I built this small demo to illustrate how the different smoothing parameters and exponential smoothing components interact. You can choose between some simulated and some real time series, as well as the option to add outliers or level shifts to the series, to explore what is the effect of different parameters in each case. The parameters can be setup either using their traditional form or the state space reformulation.
You can download the R code for this demo here.