A sweet spot is the area of the racket string bed that produces the most power, the best sound and feeling, and the least vibration when you hit on it. It is normally located between around the center of the string bed, but not directly in the middle though. The area marked in red in the illustration above is the sweet spot.
The width of the sweet spot area varies across rackets, depending on factors like racket head shape, racket technology and also design.
You should always try to hit the shuttlecock on the racket sweet spot in order to get the most efficient and accurate shot. You will know immediately when you miss the sweet spot, because the shot will be weak in power, sounds blunt and you will also most likely to feel vibration on your hand after hitting the shot.
For social or amateur players who have problems in hitting on the sweet spot consistently, get a racket with a larger sweet spot to reduce the problem. Normally an isometric racket has a larger sweet spot compared to an oval-head racket. Reducing the string tension and getting a racket with a more flexible (less stiff) shaft will also significantly widen the sweet spot. There is a trade off to this though.