While Arnel's diagram is a possible one, it has a "gotcha" that needs clarification since you claim to be novices at hardware issues. An experienced hardware engineer would know this; a novice would not.
As I mentioned, there IS such a thing as an Ethernet-enabled printer that COULD work through the router if you have enough plug ports for all of your computers and have at least one to spare. In Arnel's picture, however, you have printers connected to some computers, not the router. You can connect printers directly to the back of your computer via USB or an old-style printer port (though these days such a thing is rare.) Like Arnel's diagram shows, you CAN connect to a printer through a computer as long as you know the name of the computer, because the "connect a printer" wizard will want to know the name of that supporting computer. Given the host name, the "find a printer on the network" dialog will very quickly find such a device.
HOWEVER, if your printer is hard-wired but NOT connected to a machine that will be ON all of the time, then that printer will be unreliable. You said that one machine would be sacrificed to be the DB host. That would be a good candidate for a hardwired and shared printer to be attached.