Climbing is rated by Class and Grade
The system in use in Yosemite is called the Yosemite Decimal System (YDS).
There are many other systems: the National Climbing Classification system (NCCS),
the British System, the Austrian System, the French System including the French
(IFAS), the Soviet, West German and the European System.
This system categorizes terrain according to the techniques and equipment required in ascending that terrain, using two terms: Class (difficulty of each move) and Grade (overall length of time to climb). This system evolved originally from the Welzenbach Rating System in 1937. In the 1950’s, Royal Robbins (and Chuck Pratt?) recognized that the former system lumped too much together and did not differentiate the kind of climbing that was currently being done. For instance, there was no differentiation between a long, easy route with only a few moves of 5.7 from a short, hard that was sustained 5.7.
Class 1 - A hiking scramble to a rocky gradient; generally hands are not needed.
Class 2 - Involves some scrambling and likely use of hands; all but the most inexperienced and clumsy will not want a rope.
Class 3 - Moderate exposure may be present; simple climbing or scrambling with frequent use of hands. A rope should be available.
Class 4 - Intermediate climbing is involved and most climbers want a rope because of exposure. A fall could be serious or fatal. It begins when all beginners and most average climbers will want and should have a belay. Usually natural protection is easily found.
Class 5 - Climbing involves use of rope and natural or artificial protection by the leader to protect against a serious fall. In today's use of the terms, all "rock climbing" is Class 5 or 6.
5.0 – 5.4 - a physically fit climber can actually climb at this level with a little or no rock climbing skills, using only natural ability.
5.4 – 5.7 - Requires use of rock climbing techniques such as hand jamming and or strength.
5.7 – 5.9 - Rock climbing shoes, good skills, and some strength are usually necessary at this level.
Class 6 - Climbing that involves placing the climber's weight on the equipment itself, as opposed to using it only for protection; AKA "aid climbing.".
An interesting problem arose as climbing skills grew: what do you call something that is harder than a 5.9 but not 6.0 (aid climbing)? You could call the thing a 5.91, but that has two implied errors: (1) that a 5.91 is only slightly harder than a 5.9, and (2) it implies a level of precision that does not exist (a 5.8 is a rough estimate, but a 5.91 is very precise). The only reasonable thing was to scrap the "decimal" concept and call it a "5.10". Therefore, a 5.10 is not a more precise 5.1, but a climb harder than a 5.9; and a 5.11 is not between a 5.1 and a 5.2, but harder than a 5.10. Maybe confusing at first but it should be clear once you understand its origins.
5.10 to 5.14 - Beyond 5.9, requiring excellent skills and strength, this level requires training for climbing techniques and commitment of time to maintain that level. For reasons not known to me, each category was further broken down into four letter grades a-d (i.e 5.10a, 5.11c, etc.)
The ratings are subjective and often climbers will have a slight disagreement about the rating. However, other than regional variations, or variations from the gym to tradition ("trad") climbing, there is more agreement than disagreement, and as a matter of fact, the system works very well (ie, it serves the needs of climbers in communicating level of difficulty).
(Despite the fact that the rating would seem to apply to hiking as well as climbing, in practice it is only applied to rock climbing routes.)
Grade is the amount of time required to complete a climb. It includes some aspects of difficulty (Class) but mostly refers to length of the route.
Grade I - Single pitch climbs.
Grade II - Several pitches.
Grade III - Many pitches, three-quarters of a day.
Grade IV - Many pitches, full day, better keep moving if you don't want to bivvy.
Grade V - Most parties will spend two to three days.
Grade VI - Most parties will spend more than three days to complete the climb. (Notwithstanding this, some absolutely excellent climbers in top form with lots of speed practice on a specific route have done Grade VI's in 2 hours, 48 minutes, and 55 seconds!)