What is the difference between Proof, Brilliant Uncirculated and Bullion coins?

As Explained by the British Royal Mint:

The Royal Mint produces three types of uncirculated commemorative coin finish: Proof, Brilliant Uncirculated and Bullion. Proof coins are the highest standard of commemorative coin produced by The Royal Mint. Brilliant Uncirculated coins have a higher standard of finish than Bullion coins, without the extra finishing and detail provided on Proof coins. Bullion coins have a similar standard of finish to circulating coins.

What are Proof coins?
Royal Mint Proof coins are the highest [...]