We begin with a brief description of each class and its role in the game: Clerics are support, with some offensive and melee capabilities; Druids are like clerics, but less able in combat and more useful in the wilderness; Fighters are fighters; Paladins are fighters who are Lawful Good, and gain some clerical powers at high level; Rangers are powerful fighters who gain some spells at higher levels; magic-users are weak in battle, but have a lot of powerful and useful spells; Illusionists are like magic-users, but with a different spellcasting focus; Thieves use cunning, nimbleness and stealth; Assassins are "quiet killers of evil nature"; and Monks are trained to fight with their bare hands.
All ten of these classes have appeared in the game before, although some are altered from their original form. The following classes were introduced during the OD&D era, but haven't made the cut for AD&D: Alchemists (from The Dragon #2), Healers, Samurai and Berserkers (all three from The Dragon #3). The Bard (introduced in The Strategic Review #3) isn't listed above, but it's included in AD&D as part of an appendix. Boy, is it ever.
The opening section ends with a note that multi-class character are only limited in weapon/armor selection in regards to the Thief. Anyone else can apparently have at it, which means fighter/magic-users can cast fireballs in plate mail and fighter/clerics can use all the bladed weapons they damn well please.
From there we go to Character Classes Table I, which contains several things of note. Clerics, Druids, Fighters, Paladins, Thieves and Assassins have all started using a larger die to generate hit points (for example, Fighters have gone from using a d8 to a d10). Rangers are a special case, in that they have retained the d8 for hit points, but gained an extra die at first level: they potentially start strong, but will fall behind Fighters and Paladins as they reach higher levels. We can see that Assassins are limited to a maximum of level 14 (as before), while Druids can now advance to a max of 14th (one higher than previously) and Monks can reach a max of 17th (also one higher).
The way that hit points work for multi-classed characters is detailed below this table: basically, when you gain a level in a class you roll the appropriate die (or dice if you're advancing in multiple classes at once), add your Constitution modifier, and divide the total by your own total number of classes (i.e. a fighter/magic-user/thief would divide his total by three, even when advancing only in one or two classes). My notes are a bit hazy on multi-classing in OD&D (probably because the game itself was hazy), but this might be the first concrete explanation of how this is supposed to work.
Character Classes Table II lists the weapons and armour that the various classes are allowed to use. It follows what's been established in OD&D pretty closely. I'm always surprised to see that Thieves aren't permitted to use any kind of bows; as far as missile weapons go, they're restricted to daggers, darts and slings. There's a note below the table that characters of under 5' height can't use a longbow or any weapon over 12' long, and that those weighing under 100 pounds can't use heavy crossbows, two-handed swords or polearms over 200 coins in weight. It's a nod to realism, but it's also the sort of fiddly detail that gets forgotten during play. I think 3rd edition does it better, with the use of size categories and different-sized weapons. It's easier to remember at the table.
Of more interest are the two categories at the end of this chart: which classes are permitted to use flaming oil and poison? In the case of flaming oil, anyone can use it except for monks. I'm not sure why monks would be so averse to it when all the other classes are fine, but they do have the handy "derived from a foreign culture" origin to fall back on. I can always give them a bullshit "code of honour" of the sort that martial artists always get slapped with.
Poisons are a bit trickier. Assassins are the only class that are definitely allowed the use of poison. Paladins are strictly forbidden, as are non-evil Clerics. For everyone else, its use and availability is determined by the DM. A wise decision for such a potential game-breaker.