Many bartenders call bitters the salt and pepper of cocktails, but I think of them as the spice rack. Bitters have the ability to heighten certain flavors in booze and mixers, while also softening others. They even have the power to bring slightly disparate flavors into harmony. Much like seasoning a soup, think of the flavors in your cocktail you want to play up or compliment when reaching for a bottle of bitters.
Bitters are a mixture of botanicals, roots, and spices steeped in a spirit base. Generally, only a few dashes are called for because of their concentrated flavor.