Connect an apple LED cinema display to a new mac mini

Depending on the connectors, available adapters, & vintage of display these may vary.

• Displays - Official Apple Support

https://support.apple.com/displays

Note additional links to other articles immediately below the main one in above page.

Those cover some of several adapter or connection variables. ~ Such as this:

• Get help with video issues on external displays connected to your Mac - Apple Support

If the display is not Thunderbolt, you need a USB-C to Mini DisplayPort adapter.

https://www.uptab.com/uptab-usb-c-type-c-to-mini-displayport-adapter-4k-60hz-silver/

Don't forget to plug the display's USB cable into the computer.