What is the link between the relay used to determine the transistor used? Is the relay merely for powering on the unor3 board?
Your use of language here is confusing. Relays don't power anything. The are electrically operated switches.
If you wish to turn on and off a 12V "thing" with your Uno, then the Uno would need to operate a relay. It probably needs a smallish transistor to give it enough oomph to turn on the relay coils. Then the 12V goes through the relay contacts, not touching the Uno in any way. You would need a diode across the relay coils in this case. Google "arduino relay" or similar to get lots of ideas about how to do that (over 1.5 million hits).
You can also get solid-state relays (SSRs) from eBay and other places. These can be operated directly from a digital signal, and can switch quite large amounts of current / voltage.