The desktop is wired to the router, and accordingly the router is wired to my internet source. However, the desktop is Windows XP, and I didn't know if I needed to download something to first bring the desktop up to date. In other words, XP is an older system, and I don't know if this is a concern for which I might have to somehow compensate.
The laptop connects to the router wirelessly, and works well.
I would attempt to use the Brother software to connect to the router and install the printer as a network printer as you say, but I would rather await your instruction given the details I just mentioned, so that I don't screw anything up.