Enemy Chests
This type of chest is the most common loot cache in destiny 2 and will be guarded by a particular kind of enemy which will depend if it's a Fallen, Vex, or Carbal stash. These chests will spawn at random intervals and whenever you are in a region that contains enemies it is almost certain there will be an enemy chest nearby to collect. Enemy chests will also spawn as a reward when you successfully complete a Patrol.
Elite and Boss Chests
When you travel through Public Sectors there is always the possibility of encountering Major and Ultra enemies. Whether it is a random occourance or part of a secret event there is a chance they will drop an Elite Chest when you defeat them. Bosses will also drop chests which will contain items that are better than those dropped by lesser enemies.
Lost Sector Chests
Whenever you clear out a Lost Sector you will be given this chest as a reward. To open it will require you to obtain a code which will be given to you when you defeat an Elite foe that will be located somewhere in the sector. After you have looted the chest you can continue through the sector defeating more enemies and then return to it to get more items. You can keep repeating this process but the more times you return the poorer the quality of loot will be. These chests will reset once a week.
Region Chests
These chests spawns in a set location somewhere off the beaten path out of sight and will usually contain a higher quality of items than those found in an Enemy Chest. If you can find one when you first enter a region it will help you to get a good flow of improved Weapons and Armour.
Treasure Chests
Once you have completed the game's main set of story missions Cayde-6 will sell you a Treasure Map for 3,500 glimmer at the start of each weekly reset. Each map will have clues which when solved will lead you to the location of a Treasure Chest. Treasure Maps rotate each week with the weekly reset and each chest will have different rewards so it is advised that you buy and open the chests during that week to make sure you get their rewards.