You're at a train terminal looking for the right line to stand in to get on the train from Upstate NY to Penn Station in NYC. You've settled into what you think is the right line, but you're still not sure because it's so crowded and chaotic. Not wanting to wait in the wrong line, you turn to the person closest to you and ask them where they're going. Penn Station, says the stranger, blithely.
You decide to get some second opinions. You turn to the second closest person and the third closest person and ask them separately: Penn Station and Nova Scotia respectively. The general consensus seems to be that you're in the right line, and that's good enough for you.
If you've understood the preceding interaction, you already understand the idea behind k-Nearest Neighbors (k-NN hereafter) on a fundamental level. In particular, you've just performed k-NN, where k=3. Had you just stopped at the first person, you would have performed k-NN, where k=1.
So, k-NN is a classification technique...