What is the term for a poem that mourns the loss of a loved one?

The elegy is a form of poetry in which the poet or speaker expresses grief, sadness, or loss. The elegy began as an ancient Greek metrical form and is traditionally written in response to the death of a person or group. Though similar in function, the elegy is distinct from the epitaph, ode, and eulogy: the epitaph is very brief; the ode solely exalts; and the eulogy is most often written in formal prose.

Source: Academy of American Poets