Bifurcation theory is about how dynamical systems change their behavior as some parameter of the system is changed. You can think of the parameters as being knobs that give you control over some aspect of the system. Bifurcation theory then tells you how the system changes as you turn the knob. Many man-made systems have been engineered to have very predictable behavior as you turn the knobs that control them. For example, a volume control knob shouldn't give you any surprises: turn it up a little, the volume goes up a little -- turn it up a lot, the volume goes up a lot. Another predictable, but slightly more complicated, control is a two state light switch. Unlike the continuous control light switches, they have a point where there's a sharp transition when turning it on or off. You may also have noticed that this point changes depending on whether you're going from off to on, or on to off - this is a nonlinear phenomenon called hysteresis (related to psychiatric hysterical behavior).
One way to study dynamical systems is through the state space. This allows for a geometric interpretation of what the system is doing and gives more insight into what the system will do than you'd get from just looking at the numbers.
This technique also allows us to visulaize various changes of behavior of systems.
Strogatz, S., "Nonlinear Dynamics and Chaos: With Applications to Physics, Biology,
Chemistry and Engineering," Reading, MA, Addison-Wesley Publishing Company,
1994.