Here's a comprehensive list of everything in the game that increases the item drop rate from monsters, excepting familiars, which are comprehensively listed at familiars by type. To see various maximum levels of item drops, see Maximizing Your Item Drops.
Increased drop rates are treated as a multiplier to the base drop rate. If you have multiple item drop rate increases, the percentages are added together to arrive at a multiplier. For example, if a player has a 20 lb. Baby Gravy Fairy (50%), Mad Looting Skillz (20%), and a Mr. Accessory Jr. (25%), then the item drop multiplier will be 50% + 20% + 25% = 95%. To find your final drop rate, the game computes the "final drop rate" as
When you finish off a monster the game rolls a uniform random number between 0 and 1. If the number it rolls is less than the final drop rate, you get the item. So, in this example let's say we are farming for chef's hats which have a base drop rate of about 33.5%. Using the equipment above your final drop rate would be
When farming for more common items, a high multiplier may push the drop rate above 100%. This is just treated as 100%. Note additionally that some items only drop if certain conditions are fulfilled.
Here is a table with further examples.
Some mechanics have an item drop bonus that depends on some other character attribute. Penalties for water-based areas (and ways to cancel such penalties) are listed at underwater adventuring.