I want to buy a replacement, but don't know what to search for.
That is a 2-circuit medium base fixture socket. It is designed to accept a 3-way bulb, which is a dual filament bulb that provides 3 levels of brightness. It has a "turn knob" type switch (or maybe a push switch, it is hard to tell from the picture).
Your picture does not illustrate what type of socket holder it came from. There are various manufacturers and configurations and your picture shows the inner fixture components. You may have to buy a complete fixture socket rather than just those "guts" and be advised they are not all identical.