The most common 555 configurations are that of a Astable and Monostable Multivibrator. This shows how a 555 may be used as a Bistable Multivibrator. While a 555 has a Set/Reset flip flop built in (another type of Bistable Multivibrator) this circuit uses the concept of hysteresis to accomplish the same thing.
If you use the red wire shown on the illustration as a toggle the LEDs will flip states, and stay that way until the circuit is toggled again. It will work over the entire power supply...