Webster's defines connotation as "the suggesting of a meaning by a word beyond the thing it explicitly describes." In other words, connotation differs from denotation, the literal meaning of a word, because there are secondary associations that are carried with some words. So a negative connotation is a word that carries a bad emotion or association with it.
Poets make heavy use of connotation. They choose to place certain words in specific areas because the negative or positive connotation carried by a certain word might help the poet further the point they are making. It is also used to create mood and set tone throughout a poem.
Context determines a word's connotation. For example, most people would expect a car engine to be "greasy," but if the word is used to describe a person or food, it is usually viewed in a negative light.
In the second line of Wilfred Owen's poem "Dulce et Decorum Est," the poet uses the phrase "coughing like hags" to describe soldiers during the First World War. The word "hags" carries a negative connotation of malevolence. The word is also often associated with wizened women, so it gives the reader a good idea of the state of health of the soldiers in the poem.