Connecting my brother printer to my computer with an ethernet cord

No you don't want to connect the printer directly to your Mac with the Ethernet cable. You would want to connect the printer directly to your router or other network switch using the Ethernet cable or the printer's WiFi (if supported).

It seems Brother only supplies macOS drivers for your printer for up to macOS 10.15 Catalina:

https://support.brother.com/g/b/downloadtop.aspx?c=us&lang=en&prod=mfc7440n_all

It is possible that macOS may include CUPS or GutenPrint drivers for your printer or the general printer series. When adding the printer through the macOS "Printer & Scanners" System Preferences look for a Gutenprint driver by clicking the drivers drop down list box. I'm not sure if CUPS drivers are still shown when adding a printer these days with later versions of macOS. If the CUPS drivers are not listed, then it is possible you may need to resort to using the CUPS browser interface to configure the CUPS driver (if available), but first you would need to enable the CUPS web interface by using the Terminal command line:

cupsctl  WebInterface=yes

After which you can enter the following URL in a web browser to access the CUPS web interface where you can attempt to locate a compatible driver for your printer or printer series:

localhost:631/admin

When presented with a login prompt, just enter your macOS admin user name and password to authenticate. Unfortunately not all printers have a CUPS supported driver.

When you purchase a new printer you should make sure the printer supports AirPrint so that you don't have to worry about needing any third party drivers although many printers may only work with AirPrint over WiFi even though it should in theory also be supported over Ethernet as well:

About AirPrint - Apple Support

AirPrint over Ethernet???? - Apple Community

Hello, BMacg1rl.

Let's start off by checking the helpful steps in this article: Add a printer to your printer list so you can use it on Mac - Apple Support. This details the ways you can connect a printer to a Mac: wirelessly, USB, networked, through its IP address or Bluetooth.

Let us know if we can help with anything else at all or if this doesn't settle your issue. Thanks and have a fabulous day!