These are powerful yellow class Weapons and Armour that have unique stats and perks such as increased agility and damage recovery. Only one exotic weapon or armour can be available on each side of the load-out screen at any one time.
Exotics in destiny 2 are dropped more regularly than in the original game and you should be able to pick up your first one up from within the first few hours of gameplay if you have not already done so from an activity drop. Any Exotics you pick up can be equipped straight away provided you have the required level. To obtain the most powerful Exotics will involve unlocking and completing specific quests.
Listed below are the various ways that Exotics can be sourced:
Specific quests and missions
Random rewards from Public Events, chests, Crucible matches and other activities
Decrypting exotic Engrams
Special vendors (such as Xur)
Bright Engrams (for cosmetic exotic items)