In extremely vague, rough, oversimplified and imprecise terms, wall-crossing phenomena can be described as follows.

Given some linear category $\mathcal{C}$ (vector bundles, coherent sheaves, etc.), from a physicist's point of view it is quite natural to consider the *space $\mathcal{M}_{\mathcal{C}}$ of all objects of $\mathcal{C}$*, and to take averages of naturally defined quantities on $\mathcal{M}_{\mathcal{C}}$, obtaining characteristic numbers $I_\mathcal{C}$ (I've said I'm really oversimplifying things).

To a mathematician, the space $\mathcal{M}_{\mathcal{C}}$ makes no sense in the realm of manifolds or schemes: it is a kind of derived object known as (higer) stack. However, things are quite under control if $\mathcal{M}$ is given a stability condition, i.e., a notion of stable objects. Now, the idea of Tom Bridgeland (derived from Michael Douglas investigations on D-branes) is that there is not a distinguished stability condition, but a whole space $\Sigma$ of stability conditions (which turns out to be a topological manifold). For any $P$ in $\Sigma$ one can then consider the moduli space $\mathcal{M}_P$ of $P$-stable objects, and this leads to a formalization of the physicist's space $\mathcal{M}_\mathcal{C}$ as a bundle of moduli spaces over the space $\Sigma$ of stability conditions: the fiber over $P$ is $\mathcal{M}_P$.

Now, the geometry of $\mathcal{M}_P$ can abruptly change as $P$ moves in $\Sigma$. This is easily explained: if $P$ and $Q$ are two distinct points in $\Sigma$, there could be some object $E$ in $\mathcal{C}$ which is $P$-stable but not $Q$ stable. So in the moduli space $\mathcal{M}_P$ there will be a point representing $E$, but in $\mathcal{M}_Q$ there will be not such a point. Conversely there will be some object $F$ which is $Q$-stable but not $P$-stable. So, if we join $P$ and $Q$ by a path and follow $E$ along this path, then we see that at some point along the path $E$ must become unstable and so disappear from the moduli space of stable objects and be substituted by $F$. And this happens for each path between $P$ and $Q$: there's a *wall* separating the region where $E$ is stable and $F$ is unstable from the region where $E$ is unstable and $F$ is stable.

As we cross the wall the geometry of the moduli space abruptly changes. However this change in geometry is expected to be quite regular: more precisely it is expected to be a Mukai flop. This is too, quite easy to imagine from abstract nonsense (proving it formally is another kind of affair...). Namely, for a fixed wall, one expects that there is a whole linear space of objects which become unstable on the wall. And since we are considering objects up to isomorphisms, and we are considering linear categories, nonzero scalars will act as isomorphisms and so will have to be quotiented out. This leaves us with a projective space of objects that gets unstabilized by the wall, and so disappears as we cross the wall. But for what we just said above, there will be also a projective space of objects appearing as we cross the wall. So crossing the wall, there is a $\mathbb{P}^n$ replaced by another $\mathbb{P}^n$, and this is, very roughtly and basically, the idea of a Mukai flop. The idea of wall crossing transition of moduli spaces by flops goes back at least to Thaddeus' influential work on the Verlinde formula, studying the dependence of geometric invariant theory quotients on the choice of stability (linearization of the action), and similar wall crossings were important e.g. in Donaldson theory.

Let us now come back again to the invariants $I_\mathcal{C}$. From the mathematician's perspective we have not a single invariant, but a quantitu $I_P$ for any stability condition $P$. What happens when we cross a wall? the spaces $\mathcal{M}_P$ and $\mathcal{M}_Q$ are geometrically different, so the quantities $I_P$ and $I_Q$ are a priori completely unrelated. On the other hand, the physicist's ill-defined quantity $I_\mathcal{C}$ was defined before stability conditions, so it *does not knows* about the wall! this means that the well-defined quantities $I_P$
and $I_Q$ must coincide (or better be canonicaly related): in colloquial terms, *the invariant $I$ crosses the wall*, and the relation between $I_P$ and$I_Q$ ia a *wall-crossing phenomenon*.

Actually, this is an huge oversimplification even of the physiscist's intuition. A better description of the physicist's point of view is the following (see the excellent answer by Andy Neitzke below for details). The relevant quantity one is interested in does not originally live on $\mathcal{M}$, but in a larger space of superconformal field theories. And it depends on a paramter $t$. Let us denote this quantity by the symbol $\Omega(t)$. As $t$ varies into the parameter space of SCFTs, the quantity $\Omega(t)$ varies continuosly. On the other hand, as said above, $\Omega(t)$ is the average of some quantity $J(t)$, and this average can be computed by localization formulae as suitable integrals on the space of critical points of $J(t)$ (the spaces of vacua of the theory). It turns out that at least certain values of $t$ can be identified with stability conditions on $\mathcal{M}$; if $t=t_P$, this identifies the space of critical points of $J(t_P)$ with the moduli space $\mathcal{M}_P$ and the localization formulae relate in a precise way $\Omega(t_P)$ with $I(P)$. Now, the geometry of the set of critical points of $J(t)$ may abruptly change as the parameter $t$ varies (this is nothing strange: this happens already for families of smooth functions $f_t:\mathbb{R}\to\mathbb{R}$, as every first year calculus student knows). So the formulae for $I_P$ and $I_Q$ will be quite different if there's a wall between $P$ and $Q$. On the other hand, they are related in a precise way to $\Omega(t_P)$ and $\Omega(t_Q)$, and these quantities change in a well-behaved way as $t$ goes from $t_P$ to $t_Q$. Reading this good behaviour at the level of $I_P$ and $I_Q$ gives the wall-crossing formulae.