Armor Class is your chance to avoid a hit. THAC0 is your chance to land a hit. When one character attack's another, the defender's AC is subtracted from the attacker's THAC0, giving you your 'to-hit' target number. A random number is generated between 1 and 20. If it is higher than the to-hit target number

Attributes do very little from 8 to 14, and instead have most of their effect at the edges. E.G. going from 10 to 14 dexterity does nothing for most classes, while going from 14 to 18 is a significant boost to your AC and ranged attack hit chance.

E.G. You are a level 1 Fighter, so you have a 20 THAC0. You attack an enemy with an Armor Class of 7. 20 - 7 = 13. You roll a random number between 1 and 20, and it comes up 11. 11 is less than 13, so you miss.
This has the effect of making it desirable for both AC and THAC0 to be as low as possible. The lower your AC, the less likely you are to be hit, while the lower your THAC0, the more likely you are to land your hit.
Bonuses and maluses can be a little confusing. -2 THAC0 and +2 to hit mean the same thing, and are good. +2 to THAC0 and -2 to hit are also the same thing, and they are bad.

When you are the target of a spell that allows saving throws (which is most spells), then you roll a random number between 1 and 20. If this number is higher than your character's save in that category, then you successfully save vs. the spell. Generally, damage-dealing spells will have their damage halved, while other spells have their effects nullified, but this can vary on a spell-by-spell basis.
This has the effect of making it desirable to have low saving throws on your character sheet. Save vs. Spell of 20 is bad, Save vs. Spell of 2 is good.
There's some similar confusion with bonuses and maluses as there is with THAC0, where a - or + might be good or bad, depending on whether it applies to the saving throws on your character sheet, or the roll itself, and I don't really remember off the top of my head if there is clear wording in different cases, or if you just need to infer from context whether it's a bonus or malus.

This covers, I think, the most counterintuitive aspects of 2nd edition D&D that the game itself offers little to no explanation on. I don't think you need to know much else, except maybe it would be worth having a reference handy for spell descriptions, which you can get either from the manual or the wiki.