British Royal Mint's brief explanation of bullion, brilliant, and proof finishes. Also, the video shows some good footage of coins being minted, dies being touched up, and dies being polished. Excellent short video.
Note that the mint strikes proofs 6 times, brilliant finish coins 2 times, and bullion 1 time. Also, note the rotary feeder finger mechanism used to feed the planchets into the press for the bullion coins--if that gets out of sync you will have a "struck feeder finger."