Capacitor Delay Circuit
When you first push the button with the capacitor discharged, a current flows through the capacitor, the resistor and the LED, and the LED lights up.
That current charges the capacitor, and after about 0.1 s it will be fully charged, no more current will flow, and the LED will be off.
The capacitor will have no way to discharge, except for its internal loss, which can require a lot of time.
The next time you push the button, the capacitor is still charged, and no current flows.
So the behaviour that you are experiencing is exactly what you might expect form the circuit that you realized.

If you put the capacitor in parallel of the LED, you will see the LED remain on for a brief period of time after you release the button, and turn on whth a little delay when you push it.

If you want delays of approximately 1 second, you need at least a 1000 uF capacitor.