Crepant stands for *non-discrepant*.  It's frequently applied to resolutions of singularitie sor birational maps (but can be applied more generally).

Let's start with the birational case, since that's where the history is.
If $f : X \to Y$ is birational, and $K_Y$ is $\mathbb{Q}$-Cartier, then $f^*(K_Y)$ makes sense.  In particular, if $nK_Y$ is Cartier, then $f^*(K_Y) = \frac{1}{n} f^*(nK_Y)$ by definition.

Write $K_X - f^* K_Y = \sum a_i E_i$ where we pick $K_X$ and $K_Y$ which agree where $f$ is an isomorphism.  The $\sum a_i E_i$ is then independent of choices.  

Now, the numbers $a_i$ are called *discrepancies*.  If there are no discrepancies (ie, all the $a_i$ are zero (for example, if $f$ is a small map), then the map is called *crepant*.  Of course, all $a_i = 0$ if and only if $K_X = f^* K_Y$.

Of course, if the pullback of $K_X$ is $K_Y$, then this can be applied to many things.  The existence of a crepant resolution of singularities, also shows up in many contexts.