Chests are units of the game that contain items common to very rare ☆6 items. They drop from treasure chests or from enemies at high and low rates. In some areas, chest ranks don't really matter. Gold/Platinum chest enhancing dresses are often used to increase the chance of chest drops.
Wooden chests contain very common ☆1 items. Common items drop so much to the point where you have 1000+ of that item.
Red chests are common chess containing common ☆2 items. Like wooden chests they drop from enemies every now and then.
Gold chests can sometimes be both common, uncommon, rare on most areas. The common ones hold valuable common items that appear to be more fancier because of the gold chest. Uncommon ones are usually dropped by enemies, and they contain that enemies' themed item. The rare ones reveal more valuable items like the platinum chest would. In general mid to higher rank they drop frequently. Gold chests are also obtainable by Gambler's slot ability.
Platinum chests were introduced in 3.0 updates. They are usually very rare to get and hold very exclusive items. They contain items such as Servants, Weapon crafts, and more rare items that gold chests don't contain.
Blue chests are special chests that are obtainable via Phantom Thief's ability only. They contain horoscope themed items that change occasionally, and exp. feed shards that can save you a lot of coins on an upgrade.
Material Bags contain items necessary for weapon crafting. They are commonly obtainable in areas rank 11-15 and fairly uncommon in other areas.
Jewel chest are special pink chests that often contain accessories commonly related to the enemy that dropped it.