Once you've completed the main story campaign, you can visit the Chaos Chamber, which you can read all about in Dev Diary #5 - Challenge the Chaos Chamber. Provided you've got the skills and gear necessary to complete your first Chaos Chamber run, you'll unlock the ability to really test your mettle with Chaos Levels, where you'll find greater challenges and even greater rewards.
Chaos Levels can be easily toggled on or off, and can be adjusted to the Chaos Level difficulty you're feeling up to. With each successfully completed run you can increase the Chaos Level, which means every enemy you encounter gains proportionately increased health and damage. Defeat them and you'll earn better loot drops along with more experience and gold.
At launch, the max Chaos Level will be 20—and the cap is planned to increase further via post-launch updates. Increasing your Chaos Level is no easy feat, but you can take a stab at it via the Chaos Run option in the Chaos Chamber. Chaos Runs are the same for all players and have a set, highly demanding difficulty that acts as a skill and gear check before you can get to the next tier.
Speaking of gear checks, playing with Chaos Levels enabled means you have a chance to see Chaos Tier gear, which is way more powerful than your average piece of loot. Items of any rarity have the chance to drop with added modifiers representing higher Chaos Levels while playing with Chaos Levels active, the first of which is Chaotic.
You won't have to re-farm your favorite items every time you bump up to the next Chaos Level, however. The Chaos Tier items you find should last at least a couple of Chaos Levels if they synergize well with your build. For those Fatemakers looking to push their character builds to the absolute limit of mix-maxed power, acquiring better gear in increasing Chaos Levels will keep you continuously challenged as the difficulty and rewards scale higher and higher.