Which medical term relates to the expulsion of stomach contents through the mouth?

The term that describes the expulsion of stomach contents through the mouth is "emesis." This medical term specifically refers to the act of vomiting, which involves the forcible ejection of stomach contents via the mouth due to various causes such as illness, stomach irritation, or other medical conditions.

Understanding this term is crucial in medical contexts, as it can indicate different underlying issues in patients, such as gastrointestinal problems or reactions to medications. The other terms listed relate to different medical conditions. For instance, epistaxis refers to nosebleeds, dysuria pertains to painful urination, and erythema describes redness of the skin, often due to inflammation or infection. Hence, emesis is clearly the term that fits the definition given in the question.