Choosing the right distance is not an elementary task. When we want to make a cluster analysis on a data set, different results could appear using different distances, so it's very important to be careful in which distance to choose because we can make a false good artefact that capture well the variability, but actually without sense in our problem.
The Euclidean distance is appropriate when I have continuous numerical variables and I want to reflect absolute distances. This distance takes into account every variable and doesn’t remove redundancies, so if I had three variables that explain the same (are correlated), I would weight this effect by three. Moreover, this distance is not scale invariant, so generally I have to scale previously to use the distance.
Example ecology: We have different observations from many localities, of which the experts have taken samples of some microbiological, physical and chemical factors. We want to find patterns in ecosystems. These factors have a high correlation, but we know everyone is relevant, so we don’t want to remove these redundancies. We use the Euclidean distance with scaled data to avoid the effect of units.
The Mahalanobis distance is appropriate when I have continuous numerical variables and I want to reflect absolute distances, but we want to remove redundancies. If we have repeated variables, their repetitious effect will disappear.
The family Hellinger, Species Profile and Chord distance are appropriate when we want to make emphasis on differences between variables, when we want to differentiate profiles. These distances weights by total quantities of each observation, in such a way that the distances are small when variable by variable the individuals are more similar, although in absolute magnitudes was very different. Watch out! These distances reflect very well the difference between profiles, but lost the magnitude effect. They could be very useful when we have different sample sizes.
Example ecology: We want to study the fauna of many lands and we have a data matrix of an inventory of the gastropod (sampling locations in rows and species names in columns). The matrix is characterized by having many zeros and different magnitudes because some localities have some species and others have other species. We could use Hellinger distance.
Bray-Curtis is quite similar, but it’s more appropriate when we want to differentiate profiles and also take relative magnitudes into account.