One popular type of average that you have not mentioned is the trimmed mean (recommended by, for example, Wilcox, 2010) which I think of as a middle road between the mean and the median. You get the trimmed mean by first discarding $n$ % of the lower and upper part of your sample and then taking the mean of the resulting subset, where $n$ can be , for example, 10. The resulting average is generally more robust to outliers than the mean.
If your data looks normally distributed (or generally heap shaped) the mean is a good description of the general tendency of the data. If the data is skewed then often the median or a trimmed mean can be a better description of the general tendency.
Wilcox, R. R. (2010). Fundamentals of Modern Statistical Methods: Substantially Improving Power and Accuracy, Springer, 2nd Ed.