Chaos gives boons that grant a variety of bonuses after inflicting a debuff for several encounters.
Affinity
If given Nectar, Chaos will give you the Cosmic Egg.
Once you have given Chaos 6 Nectar, you will not be able to give more gifts until you complete a favor for them.
After you complete the favor, they can be gifted Ambrosia, although they will not give a Companion in return.
Chaos' affinity gauge is maxed out at 8 hearts, requiring the gifting of 6 Nectar and 2 Ambrosia total.
Favor
Chaos' favor is part of the same event as Nyx's favor, requiring you to reunite the two of them to fulfill one of the Fated List of Minor Prophecies. At some point, Chaos will reveal that they tried to reach out to Nyx, per Zagreus' advice. Later on, Nyx will instruct you to upgrade the eldest sigil in the administrative chamber to serve as a bridge to Chaos, which will cost you 3142 at the House Contractor. After purchasing the upgrade, you can find Nyx in Chaos' domain, and Chaos will remark that they are pleased to be able to reconnect with their daughter. From then on, you can give Ambrosia to Chaos.
Codex Entry
Unlocking the complete Codex Entry requires deepening Zagreus' relationship with Chaos.
Curses
If you're in the middle of a biome and trying to decide whether to accept a curse, check your current chamber number (upper right corner of the Boons screen) and be mindful of upcoming boss encounters. Bosses for the first 3 biomes appear in chambers 14, 24, and 36. The rooms immediately before them (13, 23, and 35) will always be non-combat chambers.
With the exception of Enshrouded, curses last for numbers of encounters (chambers with enemies) rather than chambers, so a 3 encounter curse accepted in chamber 10 would still be around for the boss fight in chamber 14 because of the shop or Pom in chamber 13.
Or a 4 encounter curse in chamber 8 could be around for the boss fight if you get a fountain afterwards - a fight in chamber 9, fountain in 10, fights 11 and 12, and shop/Pom in 13.
The curses which deal damage (Addled, Flayed, and Maimed) will not reduce your health below 1 HP.