I Moving water with a few riffles and small waves. Few or no obstructions.
II Small scale rapids; 2' waves; few large rocks; wide, clear obvious channels.
III Rapids with high, irregular waves; narrow passages often requiring complex maneuvering. 3' waves, some small hydraulics, some rocks and eddies. Scouting a good idea.
IV Long, difficult rapids with constricted passages often requiring precise maneuvering in very turbulent water. Powerful 4' to 6' waves, boiling eddies, dangerous rocks, hydraulics. Scouting necessary; conditions make rescue difficult.
V Extremely long, difficult, very violent rapids with highly congested routes. Many riverbed obstructions, steep drops, 6' to 8' waves, strong currents and hydraulics. Scouting absolutely necessary; significant hazard to life in the event of a mishap.
VI Difficulties of class V carried to an extreme. Nearly impossible and very dangerous; for teams of experts only, at favorable water levels and with all precautions.
Now, a few words about all this rating stuff. The rating system is an attempt to reduce the complex and infinitely varied features of all the rapids on the planet to one of six categories, and as such, it has its limitations. Knowing that a rapid is class III, for instance, does not tell you all you need to know to run it. The only real way to assess the difficulty/danger of a rapid is to learn how to read the water, and then go look at it yourself. The advice of guidebooks and of other experienced paddlers can be invaluable; but in the end, the decision to paddle or portage is a highly personal one that everyone who runs whitewater should learn to make.
There is endless discussion among paddlers about the alleged over-/under- rating of rivers in various regions of the world; your editor views most of this as pointless, as the difference between IV+ and V- on a remote wilderness river is probably irrelevant to someone who has blown their second offside roll attempt and is going swimming.
One observation about the scale: the ratings from I to IV tend to have a lot to do with the size of rapids; the ratings from IV to VI have a lot to do with risk factors. For example, the Chattooga's Woodall Shoals deserves its VI rating, even though it's technically a III -- unless you wind up in its terminal hydraulic.
A common question about upper end of the scale is "Does a successful run of a previously 'unrunnable' rapid make it a class VI, rather than 'unrunnable'?". The answer is no. Various expert paddlers with a certain cavalier attitude have managed drops such as Ohiopyle Falls on the Yough, but it is probably better for our collective health to continue to consider those places out-of-reach. Even though techniques and equipment continue to improve, bringing more rapids into the "runnable" category, we need to be very careful about devaluing ratings -- especially in rapids where there is significant danger.