Let $\Omega$ be the space of continuous paths from $\mathbb{R}$ to $\mathbb{R}^n$. By a famous result, it is known that $\Omega$ is a measure space if we equip it with the "Wiener measure" (see this link).
In a descriptive or intuitive sense, what does the Wiener measure tell us about a given path $\alpha$?
It seems that if $\alpha$ covers a 'lot of distance', then it should have larger measure than a path that covers 'a shorter distance' (though if $\alpha$ is bad enough, then maybe the idea of distance is not so meaningful). In particular, does the Wiener measure generalize the notion of arc-length?
Not at all. In fact, the Wiener measure of a singleton is just zero. The Wiener measure of a set is just the probability that a Wiener process trajectory is a member of that set. Thus for instance the set of functions which are differentiable at some point in $\mathbb{R}_+$ has Wiener measure zero. Similarly the set of monotone functions has Wiener measure zero.
Some simple examples of sets with Wiener measure in $(0,1)$ are the cylindric sets: these are the sets of the form $\{ f : f(t_1) \in I_1,f(t_2) \in I_2,\dots,f(t_n) \in I_n \}$ where $t_i \geq 0$ and $I_i$ are intervals. In fact these sets generate the Borel $\sigma$-algebra on the continuous functions, so the Wiener measure of any set of continuous functions can be realized as a limit using cylindric sets.