If you want to read about gradings for bouldering, read this page from the ROCKFAX site.
If you want to read about route gradings, and especially how to understand the "two-tiered" British one, read on. We also have a Grades comparison table for converting between different grading system used around the world.
Although bolted sport climbs tend to be given a single French grade, traditional climbs in Britain are graded using two figures, an adjectival and a technical grade. The adjectival grade is a descriptive overall grade for the climb, used to indicate how hard the climb is. The definition of 'hard' is a little vague, but is used to include things like how strenous, sustained and bold the climb is. The technical grade indicates how hard the hardest move on the route is, and is difficult to describe, except by saying that, for instance, a 5c climber will almost always manage a single 5c move.
Just to make things interesting, the adjectival grade covers two different aspects of a route, how strenuous it is and how well protected or exposed. Seperating the two is usually fairly easy once the nature of the climb is known - A hard adjective grade on a blank slab will mean bold, on an overhanging crack it will mean strenuous.
This can be illustrated by thinking about climbing at the limit of your grade. Here's three examples:
This hopefully shows that the two grades are not linked. The strong, fit climber who can master 5c moves will climb E2 5b no problem, but will fail on E1 6a. The bold climber will be high in the adjectival grades too, but on slabs and badly protected climbs instead. The talented but unfit climber who can manage 6b with a cigarette in his mouth might find E1 6a easy, but would pump out on E2 5b.
The Adjective grade does tend to increase with the technical grade, however, since technical moves can be physically harder, and for any individual a harder move technically will increase the seriousness on a bold climb. Most areas tend towards certain combinations of adjectival and technical grades, usually along the lines of VS 4c, HVS 5a, E1 5b etc.
Once you get the hang of the two grades, they do make sense, and they are more than twice as useful as a single grade. Basically, the British system avoids the problem which sometimes arises with other grades about, for instance, whether twenty overhanging US 5.10 moves make a 5.10 climb or a 5.12. (It's notable that the US system now includes a "risk" grade, usually given after the overall grade: these follow cinema gradings and are U (safe), R (more dangerous), X (groundfall potential) and then XX.... the latter meaning dangerous rather than sexy.
Yorkshire gritstone routes now have a three-tiered system in which the usual E1 5b has an added "P" grade, running from P0 (safe) to P3 (groundfall potential). This is because many Yorkshire grit formations do not have any useful gear placements to protect the crux or lower parts of the climb. To understand these, read the guidebook!