What's the name of the device I'm looking for
Power Management Controller / Power Management IC

The functionality your after would normally be part of a full system power management IC or a battery charging/management IC.

If all you need to implement is exactly what you asked for, a single FET with a low Drain &lt;-> Source voltage drop and maybe a couple resistor to set the trip point are all thats required.

If you need to smoothly handle bounce, such as the bounce that occurs when you plug in a DC barrel jack, you can use a supervisor/reset controller designed to deal with bounce and use the reset output to trigger the FET.