A qualitative change is a change in the nature of the behavior, in a defining feature of it. It's therefore deeper, more important than a quantitative change, by which it's meant that specific numerical values that characterize the system might change, but they still define roughly the same type of behavior.
Bifurcations describe qualitative, deeper changes, such as in the number and/or type of stable solutions of the system. For example, if a 1D system has a global attractor at $x=1.1$, then having this attractor at $x=1.3$ would be quantitatively different, but qualitatively the same: you still have a finite global attractor. Now, if suddenly you have two different attractors, or ten, or none, that's a qualitatively change - of the type described by a bifurcation.
The Wikipedia entry the OP already links lists many other examples.