What are the different types of averages? 
*

*I know there are three types of averaging methods in statistics: Mean, Mode and Median. Are there any other type of averaging methods that statisticians use?    

*How do I know which method is best for a particular set of data? 
 A: For the geometrically minded, there are means based on monotonic transformations of data.
The geometric mean of a random variable is defined as $$\mbox{G.M.}(X) = \exp \left( \int_{\Omega_X} \log(x)df_x \right) .$$
This is excellent at handling measures of the things that are known to grow exponentially such as income, bacterial colonies, disease progression, etc. One of the reasons log transforms are so highly favored in biostatistics is due to their ability to estimate geometric means with regression models.
The harmonic mean of a random variable is defined as $$\mbox{H.M.}(X) = \left( \int_{\Omega_X} x^{-1}df_x \right) ^{-1}.$$
This is excellent at estimating averages of rates where you have incidents, tasks, or events in a numerator and measures of person time in the denominator. For health planning or corporate mergers, you might be interested in staffing locum tenens doctors to travel between 3 acquired community hospitals to serve specific breakouts of MRSA. Traveling between places, you need to jointly average time per task across the various hospitals and their protocols. A harmonic mean tells you that.
A: *

*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. 
References
Wilcox, R. R. (2010). Fundamentals of Modern Statistical Methods: Substantially Improving Power and Accuracy, Springer, 2nd Ed.
A: With respect to choosing amongst the three types of averages you list, it is generally considered that the mean is appropriate for continuous equal-interval data, the median is for ordinal data, and the mode is for nominal data.  However, this scheme is quite limited.  See which-mean-to-use-and-when for more sophisticated thoughts on the topic.
A: Some relevant literature across the spectrum:
Muliere, Pietro, and Giovanni Parmigiani. 1993. Utility and means in the 1930s. Statistical Science 8: 421–32.
..gives a wide-ranging review of about a century's worth of thought on the subject, starting ~1920's with the axiomatic approach of Kolmogorov, Chisini's insights, through decision theory and other developments. A good and thorough academic review.
Many of the same insights are concisely available in:
de Carvalho, Michel. 2016. Mean, what do you mean? The American Statistician 70: 270.
For a well designed and presented, brief and accessible article the following is excellent (it would, for example, be ideal stimulation for talented students):
Falk, Ruma, Avital Lann, and Shmuel Zamir. 2005. Average speed bumps: Four perspectives on averaging speeds.
Chance 18: 25–32.
And for those who would like the full & dense mathematical treatment - really only for mathematicians to be honest - this two-part review would be a good start:
Grabisch, Michel, Jean-Luc Marichal, Radko Mesiar, and Endre Pap. 2011a. Aggregation functions: Means.
Information Sciences 181: 1–22.
Grabisch, Michel, Jean-Luc Marichal, Radko Mesiar, and Endre Pap. 2011b. Aggregation functions: Construction
methods, conjunctive, disjunctive and mixed classes. Information Sciences 181: 23–43.
