Roughly speaking, a bifurcation is a qualitative change in an attractor's structure as a control parameter is smoothly varied. For example, a simple equilibrium, or fixed point attractor, might give way to a periodic oscillation as the stress on a system increases. Similarly, a periodic attractor might become unstable and be replaced by a chaotic attractor.
In Benard convection, to take a real world example, heat from the surface of the earth simply conducts its way to the top of the atmosphere until the rate of heat generation at the surface of the earth gets too high. At this point heat conduction breaks down and bodily motion of the air (wind!) sets in. The atmosphere develops pairs of convection cells, one rotating left and the other rotating right.
In a dripping faucet at low pressure, drops come off the faucet with equal timing between them. As the pressure is increased the drops begin to fall with two drops falling close together, then a longer wait, then two drops falling close together again. In this case, a simple periodic process has given way to a periodic process with twice the period, a process described as "period doubling". If the flow rate of water through the faucet is increased further, often an irregular dripping is found and the behavior can become chaotic.