Latching power switch circuit
1)Before pushing the button When the P-Mos is off the voltage across the C1 is zero (Vc1 = 0) due to R1 and R2 pull ups.
When you push the the button the left side of the C1 becomes ground. The voltage across a capacitor can't change so quickly therefor the right side of the C1 should have the same voltage to keep Vc1 = 0. 
When this happens the P-Mos gate goes low and turns on the circuit and after that the N-Mos turns on and keeps the gate of the P-Mos low.

2) In order to have a hold on function you need to make a time constant, an RC. Put a large capacitor between the gate of the P-Mos and ground and also a series resistor with the button like 300k which gives you the same 3s hold on time.