Instances are where you go to fight normal monsters.
Instances are opened up by accepting a level quest, and defeating all previous instances.
Many instances have a story the first time you enter them, but you don't have to go through the story again if you return to the same instance.
You can fight the same instance as many times as you want.
Most instances consist of 3 fights in a row, each one a little harder than the last. However, at the end of most regions, there's a "boss" instance, with a single fight.
Instances cost 5 stamina, deducted when you win the first fight (or, for boss instances, the only fight). So, if you're not sure you can beat an instance, it doesn't cost you anything to try—but if you can beat the first fight and then can't beat the last two, you may waste 5 stamina.
Each fight may drop gold, xp, components used for synthesizing gear, and occasionally 10 diamonds. Boss fights also often drop level-1 honor gear (the same gear you can buy in the shop for 5081 gold/piece), which is worth another small amount of gold.
At the end of the instance, there's a three-card-monty-style roulette: three rewards are presented (small amounts of gold, honor, starfire crests, and diamonds), then shuffled randomly, then you click the one you think has the prize you want. Some players are convinced that you can watch the shuffling and find the right one, or that certain positions are more likely to have better rewards than others, but then some players are convinced that they have a system for real-life three-card monty too…
When you defeat an instance, you get 1, 2, or 3 stars, depending on how well you defeat it.
Different instances have different rules for the stars. For boss fights, it's usually a matter of either winning with your whole team alive, or winning before the end of round two, but not always.
For the first 7 chapters, the final chest requires 3 stars on each instance. For chapter 9 (yes, they skipped 8), you only need 30 stars from 20 instances, so you will probably finish the last chest before you've even opened the last instance.
When you select an instance that you've beaten before, you will be asked whether you want to fight it manually, or auto-farm it. Auto-farming repeats the same instance over and over, and you just collect the rewards at the end. The cost (5 stamina) and rewards (including both monster drops and post-instance roulette) are the same as if you'd fought manually.
It takes 6 minutes for each fight. You can speed this up for diamonds. (The cost goes down as you get closer to the end, but never goes below 5 diamonds.)
You can stop auto-farming at any point by clicking Pause. You get the rewards (and stamina cost) for any fights you'd already finished.
You can quit the game once you start auto-farming. If it finishes before you come back to the game, you should get the collect/confirm screen showing the rewards, but often you don't. Fortunately, you still get the rewards.
Most instances are very easy if you're even remotely up to par for your level. The only real exceptions are on chapter 9—especially the final boss fight.
Don't worry about trying to raise the star value immediately after you beat an instance; you can always come back later, when you're stronger. (Of course if you're 89/90 to a ruby that's a different story.)
Beyond the very early levels, the rewards for defeating instances are minuscule. Repeat an instance if you need to get the next star chest, finish a side/daily/guild quest, collect components to synthesize gear. But otherwise, don't waste your stamina here, unless you're about to hit 200 and have nothing else to spend it on. (If you haven't finished all your elites, and reset at least one chapter, and maybe even done extra mining/plundering, you have something else to spend it on.)
Yes, they skipped chapter 8. DBE's parent game has two new chapters that open up at level 80; DBE gets only the second and harder of these chapters, and you're expected to be able to beat it starting at level 81.
You should be able to get through most of the content, even with free companions, but it will be much harder than previous instances; you will have to watch the fights, plan your layout carefully, and maybe even spam a few times to win, even for some of the non-boss instances. The final boss, Yuni, is another story; she's harder than her elite counterpart, and even many spenders can't defeat her at first.
Fortunately, there's no real rush to get through these. You only need half the possible stars to get the final star chest. You can open up all three chapter-8 elites without defeating their normal counterparts (although you won't get the quest bonus until you do). So, if you're having trouble, just come back later, when you need the quest xp or the goal points.