"an idea or feeling that a word invokes in addition to its literal or primary meaning."
When we use words, there is all kinds of baggage that goes along with each of our selections. Like a kid walking in the door and throwing their backpack on the couch, we might not always welcome that luggage, and may wish it would be sloppy elsewhere, but it's inescapable. Also, it's a moving target. Tomorrow, the backpack might hit the floor of the closet, instead. English is uniquely an organic language, where words that meant one thing a decade ago might mean something else entirely now.
For an example, try these phrases on for size:
"He's my boyfriend."
"He's my paramour."
"He's my sweetheart."
They all mean the same thing, right? But you probably heard them mentally in different voices, with differing images of the person speaking the words.
Connotation is a powerful tool in our writing kit. With one word, we can evoke so much, but be careful - it can go the other way, as well. If you use the phrase "making love" you know what it means, but if you are reading, say, Zane Grey's novel Nevada, you may be struck by how he uses that exact phrase. Once upon a time, it meant flirting, courting a girl, and if you read that with the modern connotation, it can be oddly jarring!