Drops are items obtained from killing a Monster through Combat. Many important items can only be obtained as Monster drops. Because many rare drops, especially Weapons, are highly sought after, selling them on the Player Market is a good way to obtain Coins. Similarly, since obtaining drops depends on RNG, if you have trouble obtaining an important drop, many can be bought on the Player Market.
Drop Rate refers to the rate or chance of obtaining a drop. Each Monster Drop has a specific chance of being given, such as 1/5, which is determined randomly by RNG. Currently, there is no special effect to change the Drop Rate, however Shiny Monsters act as if 3 Monsters have been killed, rather than 1.