A converter (a.k.a. converter/charger) handles power and charging needs when connected to shore power:
For turning 12v into 110v see inverters.
OEM converters generally do a good job of providing 12v to the house. They can do this even if the battery is absent or dead. All the RVer has to check is if the converter Amperage rating is sufficient for present needs.
OEM converters are notoriously cheap and dumb (lacking smart charging functions). Since they are stuck at one voltage the manufacturer picks a compromise setpoint. If this is set too high or too low for your usage patterns it can damage lead-acid batteries (batterycide).
This compromise setpoint should coincide with standard float voltage (Vfloat) for your battery bank. Typically this would be 13.2v - 13.4v for converters that are always on the grid, and 13.8v for batteries that are cycled but get put back on grid charging regularly.1)
Aftermarket converters tend to be higher quality than OEM, but some are still “dumb” single-stage chargers. Higher-end aftermarket converters often have multistage charging built in, or available as an upgrade.2)
Three converter makers are generally respected on the CRVL forum. In alphabetical order:
SternWake says he leans toward Progressive Dynamics “for the ability to choose stages, holding 14.4 for 4 hours at the press of a button”3)
Dumb converters output a single voltage (see above).
It is possible to build a DIY converter from a power supply and MPPT charge controller.
If you do not need smart charging an adjustable voltage power supply could do the trick. Since by definition the batteries aren't cycling when on shore power you could adjust it to something like 13.2v and let it “float” charge over time.