This circuit could be used to toggle on and off a NE555 output by a single momentary button. Similar circuits are often used instead of mechanical bistable relays, especially when the circuit must back to the start condition by switching off the power supply.
At the start condition pins 2 and 6 are supplied with half voltage, then the output (pin 3)keeps the low level. By pressing the button Q1 turns on with a small delay, while Q2 keeps off due to the collector of Q4.
Once the capacitor is charged Q1 leads pin 2 and 6 to low voltage (about 3V) and prevents Q2 from switching on. By releasing the button the capacitor discharges. Since the output of the NE555 is now high Q3 and Q4 are off. So by pressing the button again Q2 switches on and leads pin 4 to low voltage (about 0V), then the circuit backs to the start condition.