Previously used mouse is not working on my new MacBook Air

If the mouse is NOT bluetooth - would that has the USB transmitter?

I have no idea your "previous MBA" model - that would be really helpful to determine your situation.

The nowadays MBAs are equipped with USB-C/Thunderbolt interface and you will need an adapter (USB-C --> SB-A).

No, you do not need Apple mouse..

Most mice should just work – once you figure out how to pair them (Bluetooth mice), or plug them in (wired mice, and wireless mice with proprietary USB receiver plugs).

If your mouse has extra buttons – besides the usual (left + middle + right) or (left + scroll wheel + right) – it might have come with software to help you assign functions to the extra buttons. In that case, you might need to check whether the vendor has software that is compatible with your new computer.