The WP article is pretty good, but SE is meant to be standalone, so I'll try to give my interpretation of the American system, the Yosemite Decimal System. This system is for free climbing (mountaineering, trad climbing, sport climbing, and gym climbing). It doesn't cover aid climbing or bouldering.
1 = Hiking. Example: Kilimanjaro.
2 = May use hands for balance. May include cross-country where you need to pay constant attention to footing. In the Sierra, this is often talus. Example: Mt. Langley.
3 = Scrambling, possibly using hands to pull yourself up. Most people would downclimb facing outward. Examples: Avalanche Gulch on Mt. Shasta, Whitney Mountaineers' Route.
4 = Hard scrambling, with lots of good holds. Most people wouldn't be able to downclimb facing outward.
5 = Vertical rock climbing. Most people would rappel down rather than downclimbing. Example: Mount Kenya.
The grade for a route is intended to represent the difficulty of the single hardest move. For example, people who have done the South Col of Everest say it's almost all a walk-up (class 1-2), but that the Hillary Step is class 4, so the route as a whole would be class 4.
The YDS does not attempt to represent the level of danger, strenuousness, exposure, or commitment. For example, the Ebersbacher Ledges in the north fork of Lone Pine Creek are class 1, but very exposed; if you tripped over your shoelaces and fell over the side, you'd be dead. Ratings may be for a person of typical height, or may be defined according to the perceptions of a person who is a particular height.
Ratings don't necessarily relate to whether ropes are necessary. A belay might be advisable on a class-1 ledge if it was exposed and had patches of ice. A belay might be unnecessary on a class-5 climb it it was not exposed and the whole climb was extremely short and consisted of a single hard move.
In mountaineering, you have slab climbing, crack climbing, snow, and ice, which aren't directly comparable to vertical rock climbing. It's pretty subjective to say that an ice-ax-and-crampons route like Avalanche Gulch on Shasta is the same class-3 grade as a talus and scrambling route like the Whitney Mountaineers' Route.
Within class 5, there are decimals. Originally these went from 5.0 to 5.9. These subdivisions were defined using 1950's equipment (hobnail boots, no SLCD's) in the Tahquitz Rock area, with 5.9 defined by the climb Open Book, which was first climbed by Royal Robbins in 1952 and was at that time thought to be the most difficult climb that anyone could do. Today, we have sport climbing, various improvements in equipment, and a higher-level elite of pro climbers like Adam Ondra and Chris Sharma. The scale has therefore been extended up to difficulties like 5.14 or maybe even 5.15. At these very high levels, it's difficult to confirm proposed ratings, because there are only a few people in the world who could climb both a 5.14 and a 5.15 and give a reliable opinion on which was which.
Within each 5.x grade for x=10 and above, there can be letter subdivisions abcd, so that, e.g., 5.10b is harder than 5.10a.
Ratings are extremely subjective, and local customs may cause them to be either inflated (too high) or "sandbagged" (too low) compared to the ones in the Tahquitz area, where they were defined. For example, I've managed to climb some 5.10a routes at my local climbing gym, but that absolutely does not mean that I'm climbing better than Royal Robbins; it means that (1) the ratings at my gym are inflated, and (2) I'm climbing these routes either on top-rope or leading them in an indoor sport-climbing style, which is a lot easier than a trad lead.
In the 5.x grades, it can be very hard to compare different types of climbs. For example a 5.10a in the gym could be:
A climb that's overhung the whole way, but has huge holds, requiring stamina and upper-body strength.
A climb that's vertical and requires delicate balance and flexibility.
A climb that's less than vertical but has tiny sloping holds, requiring strong hands.