I have a WD My Passport which was backing up my old computer. Can I use it with my new Mac mini M1? If so, do I need to get a new connecter cable for it? It is for USB 3 or 2.

You can connect it to the USB-A ports directly or use a USB-C to USB-A adapter to connect it to the Thunderbolt4/USB4 ports. The choice is yours. However, since it is a spinning disk hard drive, their is no speed advantage by connecting through the Thunderbolt4/USB4 ports.

Can I use it with my new Mac mini M1?

Yes, likewise, you might connect to the M1 MacMini during migration process (M1 MacMini has two USB 3 ports - no need to mingle with USB-C port).

After the restoration (I assumed you use TimeMachine as backup method) - the M1 Mac Mini will ask you: If you want to inherit the old backup.. In case you want to use this WD for the old Mac Mini, then just eject the drive and designate WD for old Mac Mini. If you want to transfer the WD drive to new M1 - you can either wipe it out as new backup drive, or inherit it and continue the backup.

You can use it.

USB-C - USB adapter

http://store.apple.com/us/product/MJ1M2AM/A/usb-c-to-usb-adapter

You can continue to use your old drive or erase and start fresh if you so desire.