Juicing is the process of extracting juice from fruit and vegetables. Juicing machines for personal use are called juicers and range in price from a few dollars to over a thousand, and drinking juice from these things is supposed to cleanse the body of non-specific toxins. The real purpose of these frequently expensive devices, though, is to empty the buyer's wallet, as drinking juice from a juicer is no healthier than eating the whole food or (by extension) drinking a smoothie made with an ordinary blender. (Those who really want or need pure liquids can just buy any inexpensive juicer or filter blender-made smoothies with a strainer.)
Advertisers make some amusing claims, such as "this juicer is more efficient than any other juicer on the market"; of course, eating the actual food is 100% efficient, and using a blender is nearly so (depending how thoroughly you scrape out the blender into your glass). They also frequently highlight the fact that you can add sweet fruits to vegetable cocktails to make them tastier. What they don't tell you is that doing this has a side effect: Since juicers put all the solid parts into one bin, when the juice is done you're left with stringy, unpalatable, orange-broccoli-apple-spinach-grapefruit-tomato-carrot leftovers, which you'll probably throw out. So much for efficiency.
 Other uses
"Juicing" is also a slang term for athletic use of performance-enhancing drugs.