The single-power-button logic of the cell phone
PMIC, per the answer from @crasic.
Any device (e.g. microcontroller) that has a zero-power or near-zero power sleep mode. For example Microchip offers parts with sleep currents in the nA range. (Just use an interrupt pin to wake back up.) Processor will control power to the rest of the circuit.
A discrete solution is a circuit that I've heard called a "suicide circuit" because the host processor can cut off its own power. However when I try to Google the term "suicide circuit" I get hard-core workout routines and commentaries about stand-up comedians who have been pushed to the edge. I guess "suicide circuit" isn't exactly standard terminology. Anyway, here is the basic concept for the circuit:








simulate this circuit &ndash; Schematic created using CircuitLab


When SW1 is pushed, Q1 gets turned on and power is supplied to PWR_IN.
Once the processor is awake (optionally with some debounce time / reads) it asserts HOLD_PWR_ON. This turns on Q2.
At this point, Q1 will remain on regardless of whether or not SW1 is still pressed or not.
...and the SW1 input can be used as a standard switch to be read by the processor.
When it's time to power down, the processor sets HOLD_PWR_ON low and cuts its own power.


This of course can be done with FETs instead of BJTs if desired. As you'd probably guess, you want devices with nice low leakage in the OFF state. And it needs a processor (or PMIC) that wakes up quickly to set HOLD_PWR_ON, to guard against to short of a press time on SW1.