This is the same diagram as Figure 2-71 from Charles Platt's book Make Electronics. It's a basic example of how a capacitor can be used to keep the red LED glowing longer: now when you press the left button, the RED will take longer to go dark. Note: the right button is to "reset" the circuit if needed. I have increased the value of the capacitor to make the delay more clear The original value of 1,000uF didn't really delay the red LED.
