Boaters all over the world have agreed upon a common rating system for rivers/rapids and this is called the International Scale of River Difficulty. The system combines actual difficulty along with consequences from a mistake. Individuals rating rivers and rapids can be very inconsistent. What is easy for a world class paddler can be downright difficult for the average intermediate boater. The standard classification descriptions do provide a number of features to help ensure consistency. When planning a river trip on a river you haven't paddled before, I highly recommend researching the AW site: AW River Database. The AW ratings have been vetted by many paddlers and are generally very consistent. Guidebooks are another great source and often provide a great deal more information on the actual streams. For a humorous version, check out the following link: AMC Whitewater River Ratings - Humor.
Another system that is used locally is the Novice, Practiced Novice, Low Intermediate, etc. rankings. This ranking system is described nicely on the following link: MCC Novice, Practiced Novice, etc. Rankings.
I have copied the AW section on River Classes here:
Class I Rapids
List of Class I thru III Rated Rapids
Fast moving water with riffles and small waves. Few obstructions, all obvious and easily missed with little training. Risk to swimmers is slight; self-rescue is easy.
Note: A great local example is the Feeder Canal
Class II Rapids: Novice
List of Class I thru III Rated Rapids
Straightforward rapids with wide, clear channels which are evident without scouting. Occasional maneuvering may be required, but rocks and medium-sized waves are easily missed by trained paddlers. Swimmers are seldom injured and group assistance, while helpful, is seldom needed. Rapids that are at the upper end of this difficulty range are designated “Class II+”.
Note: A great local example is the Z Channel and Muddy Creek
Class III: Intermediate
List of Class III Rated Rapids
Rapids with moderate, irregular waves which may be difficult to avoid and which can swamp an open canoe. Complex maneuvers in fast current and good boat control in tight passages or around ledges are often required; large waves or strainers may be present but are easily avoided. Strong eddies and powerful current effects can be found, particularly on large-volume rivers. scouting is advisable for inexperienced parties. Injuries while swimming are rare; self-rescue is usually easy but group assistance may be required to avoid long swims. Rapids that are at the lower or upper end of this difficulty range are designated “Class III-” or “Class III+” respectively.
Note: A great example of this class river is the Lower Yough at summer flows.
Class IV: Advanced
List of Class IV Rated Rapids
Intense, powerful but predictable rapids requiring precise boat handling in turbulent water. Depending on the character of the river, it may feature large, unavoidable waves and holes or constricted passages demanding fast maneuvers under pressure. A fast, reliable eddy turn may be needed to initiate maneuvers, scout rapids, or rest. Rapids may require “must” moves above dangerous hazards. Scouting may be necessary the first time down. Risk of injury to swimmers is moderate to high, and water conditions may make self-rescue difficult. Group assistance for rescue is often essential but requires practiced skills. A strong Eskimo roll is highly recommended. Rapids that are at the lower or upper end of this difficulty range are designated “Class IV-” or “Class IV+” respectively.
Note: A great example of this class river is the Cheat Canyon at 3'
Class 5: Expert
List of Class 5 Rated Rapids
Extremely long, obstructed, or very violent rapids which expose a paddler to added risk. Drops may contain** large, unavoidable waves and holes or steep, congested chutes with complex, demanding routes. Rapids may continue for long distances between pools, demanding a high level of fitness. What eddies exist may be small, turbulent, or difficult to reach. At the high end of the scale, several of these factors may be combined. Scouting is recommended but may be difficult. Swims are dangerous, and rescue is often difficult even for experts. A very reliable eskimo roll, proper equipment, extensive experience, and practiced rescue skills are essential. Because of the large range of difficulty that exists beyond Class IV, Class 5 is an open-ended, multiple-level scale designated by class 5.0, 5.1, 5.2, etc... each of these levels is an order of magnitude more difficult than the last. Example: increasing difficulty from Class 5.0 to Class 5.1 is a similar order of magnitude as increasing from Class IV to Class 5.0.
Class VI: Extreme and Exploratory Rapids
These runs have almost never been attempted and often exemplify the extremes of difficulty, unpredictability and danger. The consequences of errors are very severe and rescue may be impossible. For teams of experts only, at favorable water levels, after close personal inspection and taking all precautions. After a Class VI rapids has been run many times, its rating may be changed to an appropriate Class 5.x rating.