Non-metric distances
For a really thorough listing of various distances, metric or not, that are in use, I recommend Encyclopedia of Distances by Deza and Deza. (It is also available electronically from Springer.) For a really simple example of a family of non-metric distances, you can simply let $p$ be less than 1 for an $L_p$ distance, for example. For a family that is in heavy, actual use, you could look into various forms of weighted edit distance. There's also the cosine measure, Jeffrey divergence, partial Hausdorff distance, dynamic time warping distance, and domain-dependent distances such as the COSIMIR model, all discussed in Skopal's paper on indexing non-metric distances.