The PS5 controller doesn't just work with the PS4, so you must pick up a Bluetooth adapter. This will usually come in the form of a USB device which needs to be plugged into the PS4. The adapter then ...

The PS4 controller is usually bundled with the PS4 gaming console when you buy it. You will get the DualShock 4 wireless controller which does not need any wires to connect to the gaming console.