A metaphor for any paired motivational tools: Threat and incentive, risk and reward, bad and good, excruciating pain and sex.
The "stick" is the bad side--I will beat you with it if you do not comply. The "carrot" is the good side--I will feed it to you if you do comply. Let's avoid confusion right now and allow me to point out that "Stick and Carrot" has nothing to do with dangling a carrot in front of a donkey so he perpetually chases it. That's the proper use of the carrot--the perpetual incentive--but it completely ruins the yin and yang of the situation, and confuses the daylights out of people about what the stick is really for. If it were that cliche image from cartoons, it would be called "Carrot and Stick and String."
To be effective, the carrot needs to be a tangible reward, but the stick needs to be a tangible threat. Various theories on behavioral science talk about to what extent one should use a stick or a carrot, but YMMV: experiment a little bit.
It is important to note that the stick doesn't have to be a physical punishment, and the carrot doesn't have to be food. If your subject enjoys watching The Simpsons, the carrot could be permitting him to watch. If your subject dislikes a certain song, the stick could be playing that song over and over and over again. And if your subject is allergic to carrots, or is into S&M, it's entirely possible that your stick could be a carrot and your carrot a stick.