When you apply catastrophe theory, you basically make Taylor expansions of your equations close to a bifurcation point and study how the parameters affect the behaviour. Then, according to the mathematical structure of your expansions, called "germs", you might reach conclusions about the behaviour of your system close to the bifurcation points.
In a more general sense, as explained in Robert Gilmore's excellent book, catastrophe theory is a "program", in the sense that it attempts to study how the qualitative nature of the solutions of equations depends on the parameters. Although in its more general form catastrophe theory deals with any discontinuity, there are in practice insurmountable difficulties with the most general equations. In practice, what has been done to a some extent successfully is to study the discontinuities of simple gradient dynamical systems. One aspect that might explain why catastrophe theory fell into almost oblivion is the silly turmoil about it in the 70's. Another, and more important, is that catastrophe theory does not suit itself very adequately to quantitative prediction and calculation, like applied dynamical systems, although it can be used for that.
Without claiming any expertise, I have been a fan of this branch of mathematics since about 2005, literally devouring books and articles. Arcane as it seems, catastrophe theory is an illuminating and very deep set of ideas, very closely related to topology, chaos and dynamical systems in general. I would recommend the above mentioned book to anyone interested in such things. Interested people might want to look to the above mentioned book, and also Poston and Stewart's, that has an excellent and gentle introduction. Of course, there is the very hard to digest original: Structural stability and morphogenesis. Vladimir Arnold wrote a very good book on the subject, which might appeal better to modern mathematicians. Enjoy!