The cluster of implications that words or phrases may carry with them, as distinguished from their denotative meanings. So does the advertiser or the propagandist. The scientist and the philosopher try to stay away from connotations as much as possible because connotations can vary and be difficult to pin down or agree upon. Denotations, on the other hand, can be made explicit and exact. Thus words can seem to be made to mean no more and no less than what we define them to mean.
People turn to dictionaries to discover the agreed upon denotative meaning.
But when writers use words that have associations and emotional overtones that are harder to pin down or agree upon, they use them primarily to convey and create emotional force. I think of the connotation of a word as its aura or aroma, a kind of smell the word has. If you can tell the emotional company the word keeps, then, that smell is meant to rub off on the words around it or the subject the word is aimed at. A word might connote guy talk, for example, if it is the word guys sometimes use. It then might connote a male or masculine world.