Examples of non symmetric distances
A classic example is the circle $S^1$ with metric the length of the shortest clockwise path between $x$ and $y$, but let me say some things in general.
Lawvere once made the point that this is really a much more natural definition of a metric space, since it allows them to be interpreted as a type of enriched category. The triangle inequality then becomes a consequence of composition of morphisms, which is extremely reasonable if you think of distances in a metric space as measuring "the best way to get from $a$ to $b$": clearly the best way to get from $a$ to $b$ to $c$ is at most as good at the best way to get from $a$ to $c$, and this is precisely the (asymmetric) triangle inequality.
On the other hand, there is no reason in general that the best way to get from $a$ to $b$ has to look anything like the best way to get from $b$ to $a$. This is the sense in which the symmetry requirement is unnatural. There is also no reason in general that it should be possible to get from $a$ to $b$ at all! This is the sense in which the requirement that distances be finite is unnatural. Finally, there is no reason in general that it should not be possible to instantaneously get from $a$ to $b$ (in other words, that $a$ and $b$ be isomorphic); this is the sense in which the requirement that distances be positive-definite is unnatural.
Here is how to use that idea to generate a large class of examples. Let $(M, d)$ be a metric space and let $h : M \to \mathbb{R}$ be a function. Define a new metric $$d'(x, y) = \begin{cases} d(x, y) + h(y) - h(x) \text{ if } h(y) \ge h(x) \\\ d(x, y) \text{ otherwise} \end{cases}.$$
Intuitively, $h$ is a potential (e.g. a height if one is thinking of gravitational potential), and the new metric $d'$ penalizes you for going against the potential (e.g. going uphill).
The directed graph example given by mjqxxxx is also a good illustration of this philosophy about metric spaces.
Most distances in real life are going to be more or less asymmetric due to one-way roads, going uphill resp. downhill, different public transportation schedules, congestion, etc.
The Hausdorff distance is a symmetric version of a natural non-symmetric distance.