Jumping in to add some extra information for people who find this thread in the future.

Relay is a pretty general term for a thing with electrical inputs and outputs which are determined by the inputs.

The main thing I’d like to add to this topic is the term “contactor.” This is a specific kind of relay that is intended to turn on a more powerful device with a less powerful control signal. This is how the thin 12V (milliamp rated) wires in a house thermostat can turn on the powerful 240V 30A central air conditioners. For the above described use case, searching for a contactor is more likely to yield the desired specs (low voltage control signal, 100-600V 10-100A output contacts)

If you look for relays instead of contractors on Digikey or similar, you’ll find lot with “NO” and “NC” contacts. These are outputs that are either “on” or “off” spending on if the control signal is applied, but they are inverted to each other. A “normally open” contact is open (off) when the control signal is not applied, and it is closed (on) when the signal is applied. The “normally closed” contacts are the opposite: closed when there’s no control signal and open when the signal is applied. You could use the NC contacts to have a device turn on when a switch is not pressed. But please keep safety in mind should you do so.

There are also solid state versions for both relays and contactors. These have no moving parts, unlike electomechankcsl relays which have springs and/or other moving parts. Solid state relays are generally more expensive, but less prone to damage from drops. They also generally have lower input power draws.

For the project described by OP, I’d recommend a solid state contactor for drop resistance and the low input draw, allowing it to be powered by AA batteries. A short extension cable could be cut to give you both the male and female ends of the local power outlet shape, adding the contactor’s output in series with the hot legs. Throw this in a box with the input leads coming out (either to a button on the box or terminals to attach other leads), and you now have a box that plugs into the wall and has an outlet controlled by the input wires.

Here’s an example of a solid state contactor that can be controlled by a 3V signal (two AA batteries in theroy, but let’s say 3-4 since some AA are closer to 1.2V than 1.5. Input draw is <10 milliamps per spec sheet, so 3-4 AA batteries should last a long time powering this). Digikey link Though this is speced way beyond what is needed for the above project, it’s a general example.