For the rest of the article, I'm going to draw examples from two of my favorite boss fights: Ganon from The Legend of Zelda: Ocarina of Time and GladOS from Portal.
I am going to spoil these fights pretty thoroughly. Proceed with caution if you're seeking to avoid spoilers.
As I mentioned earlier, boss battles tend to be structured based on a series of "story" beats. I've identified eight beats that I like to use when designing boss fights. Below, I've stated the nickname I use for each beat, how it works, why it's a good thing to do, and then cited two examples of each.
What is it? This beat happens before the player even gets into the fight. Just like with pay-per-view boxing, wrestling, or MMA fights, a boss fight needs to be promoted. The player needs to be informed how awesome, dangerous, vile, etc the boss is through cutscenes, dialog, or any number of other methods. The player also needs to be trained on the skills he'll need to beat the boss.
Above all, however, the main point of this beat is to increase the intensity leading up to the boss fight itself.
Why is it a good idea? As we discussed above, a boss fight is a test of the player's skills. It is, therefore, important to train your players on how to fight the boss. Most often, this is done during the build-up to the fight.
Anticipation is also really important for the pacing of your game. If you want your boss to feel like an emotional milestone, you want to create anticipation in the player to increase intensity and to raise the emotional stakes.
Ocarina of Time has a number of great examples of the "Build-Up" beat. Ganon (the evil wizard behind everything bad in the game) makes a number of appearances in cutscenes, kidnaps princess Zelda, and otherwise makes a nuisance out of himself for most of the game.
My very favorite instance of this beat is the "Phantom Ganon" boss battle. The designers of this game are so hardcore that they created another boss battle just to train you on how to eventually defeat Ganon at the end of the game.
Part of the way through the game, the player fights a boss battle against a ghost that takes the form of Ganon. The battle teaches the player that he can reflect Ganon's yellow energy attacks back at him to stun him and the player learns he needs to shoot Ganon with his bow and arrow while he's stunned.
In an earlier boss battle, the player fights "Phantom Ganon." This boss teaches the player that the he can use his sword to reflect yellow energy attacks back at the boss.
The player needs to have two particular skills in order to defeat GladOS. The game teaches the player these mechanics in two amazing and incredibly memorable sections.
First, over the course of the game the player is given only one friend; a metal block with a heart on it named the "Companion Cube." During a particularly memorable section, the player is forced to destroy the cube by throwing it into an incinerator while GladOS taunts him.
In this emotionally-charged segment, the player must pick up his Companion Cube (pictured bottom-left) and throw it into the incinerator. This trains him on the incinerator mechanic so he can use it to defeat GladOS.
Second, the player is confronted with an indestructible robot that fires missiles at him. As he works his way through the segment around this enemy, he finds he can use the robot's rockets to get past obstacles and destroy things in his path.
The player must use the missile-launching robot (pictured center) to destroy obstacles in his path. This trains him on the mechanic so he can use it to defeat GladOS.
Finally, GladOS repeatedly promises you cake. But it's a lie! A LIE! This may seem trivial, but I REALLY wanted that cake.