Yes, you'll need a relay, transistor, or MOSFET.
The regular Arduino works at 5V so that's the input/output voltage. (There are lower voltage versions but there are no microcontroller chips that run on higher voltage.)
There is a built-in voltage regulator so the board can be powered with 12V, with the onboard electronics running at 5V.
You can easily handle higher input voltages with a voltage divider (2 resistors) or an over-voltage protection circuit (one or 2 diodes and a resistor).
The Arduino can't directly supply enough current to directly-drive a relay coil but you an buy relay boards with a built-in driver. Or, there are solid state relays. Of course, whatever you choose needs to handle the current (Amperage) of whatever you're switching. (Any relay, MOSFET, or transistor will be able to handle the 12V.)