For most of the "boss" monsters encountered in normal instances (usually at the end of a region), there is a corresponding elite version.
The elite boss will appear once you've reached the appropriate level and defeated all earlier elites.
You will get a quest to fight the elite boss once you've defeated the normal boss. Often, this happens before you can reach the elite boss (because you're stuck on an earlier elite). Occasionally, it happens after you've already defeated the elite boss (because you're stuck on the normal instance).
Unlike normal instances, you can only beat an elite once per day—except that you can pay 50 diamonds to reset a whole page of elites.
Elites have similar drops to normal instances, except that they will also give you a plan for gear of the appropriate level (which you can sell if you don't need it), often a vitality potion for aeon practice, and 10-diamond drops may be more common. Elites have the same three-card monty afterward as normal instances.
Each elite chapter has star chests across the bottom. In addition to diamonds and gems, this is where you find the contracts for many of the reputation companions.
Auto-farming works differently for elites than for normal instances. Instead of picking an instance and repeating it multiple times, you pick a chapter and attack each elite in that chapter (skipping over any you've already beaten today).
Other than that, it works the same: 6 minutes per fight, same speedup costs, normal stamina use, same rewards as if you'd fought manually.
For dragons and other solo fights, make sure to put a single-attack skill on your main (the +lethal/+crit one usually makes sense), or remove any Havoc so you don't skill.
Getting 12 stars on chapter 6 gets you Lulu, the last free companion in the game. Most people do this by getting 3-3-3-2-2-2 or 3-2-3-2-3-2 stars. Getting that third star on Vanessa (or Vaz'jar) is usually the hardest part.
Cerberus is a single dragon.
He has a very strong skill attack, so the key to beating him is to kill him before his rage reaches 100.
When you first reach him, the easiest way to do this is to leave your lowest-damage companion home, leaving his slot empty or filling it with a non-attacking companion like Leo. That means his rage will only be 80 on his second turn, giving you a whole extra round to kill him.
Later, you should be able to kill him before his second turn, so you can stop doing that. Use Might, with your highest-damage comp in slot 1 (so he'll get his second attack before Cerberus's). (It doesn't have to be Might, just anything where slot 1 isn't in the center row, so your killer isn't killed by Cerberus's first-round attack.) That should get you the 3 stars.
Vanessa herself is easy to kill, but she's hiding behind a guardian, and if you can't kill her by the second round, she will skill blast you badly.
If you have two good counterers (e.g., Kelvin and a berserker main), use Ferocity, with the two of them on top. Her normal attack is a row attack, and if you can double-counter it, she should be dead, or close enough to it that Andre or another all-attacker can finish her off. (It should be possible to get Andre to take two hits and skill even with only a level-6 havoc.)
If you have a good enough assassin (Kasumi or Fred), or maybe the back-attack skill on your main, you can rely on that instead (or in addition to a single counter).
The only other tricky part is that the bottom row has back-attackers, so you probably want to put your bottom row in the reverse order you'd usually use.
Once you have two level-8 Havocs, this becomes a lot easier. Or just one havoc-8 and a strong enough skill attacker (main, Ying Chen, etc.).
Another single monster, but this one is easy enough that most people 3-star him as soon as they see him, using whatever formation they had on for Vanessa.
Much like Vanessa, but Cyclop has no row-attack to double-counter.
So, you only get one counter, which means you probably need more skill attackers.
If you can tranquilize him with your main skill, your other skill attackers can wait until the second round. But just Andre and Bareth may not be enough skill damage to kill him; you may need Martina's cross attack as well. Evasion with main, Bareth, Andre, Kelvin, Martina works for that.
You will need Martina to be strong enough to finish off the guardian by the next turn; otherwise Kelvin ends up wasting all of his attacks on a phys-immune target, and you can easily end up dying to the rest of the team even though you've killed the boss.
Later, you'll have enough Havocs to get everyone off, or Ying Chen to tranq so your main can focus on damage, or just a lot more damage from your main or Bareth. That's when you'll get three stars.
At least for trying to 3-star Vaz'jar later, the same Ferocity layout that worked for Vanessa seems to work well.
Yes, they skipped chapter 7. There's a whole page of content (plus the corresponding page of normal instances) that they apparently forgot to copy over from DB to DBE, and DBE gets content designed for levels 91-100 starting at level 81. Yay?
With a strong enough berserker or swordsman main with a pretty good skill attack, his column skill, plus one counter, plus one normal attack should be good enough to take out two enemies almost on his own. Which is good, because most of the rest of your team isn't going to survive that long.
This one is surprisingly a lot easier than normal Yuni.
Each of the three is immune to one kind of attack, so you have to arrange your team carefully (Martina in the middle row, Kelvin on the bottom, etc.). They also have pretty high resilience, so you can't rely on criticals to win.
But otherwise, they have lower defense than normal Yuni, and aren't debuff immune, so it's not hard to take them down.