Just for the heck of it, I thought I'd post a bunch of things I've collected over the years.
AI warfare unit bonuses: At Monarch and above, the AI civs get extra offensive and defensive units when they start the game. These are the best units that the Civ could build at that time; usually these will be normal units (warrior, with archer or spearman if the civ knows the appropriate tech.) These can be unique units if those can be created at the start of a game by that civ.
AI non-military unit bonuses: The AI gets an extra settler at Deity level, and either one or two extra workers at the top two levels, respectively. (Note that if you look this up in the Civ3XEdit, the help file there helpfully has the actual unit type reversed.) In my view, the extra settler is one of the greatest bonuses the AI has at the Deity difficulty level. This gives the AI a huge jump at the start of the game, where a lot of games are decided.
AI unit support bonuses: This gives the AI civs the ability to have a lot of units without paying for them. At Deity level it's 16 free units to start and 4 extra for each city it has. A large percentage of the time the AI never even has upkeep costs at this level, even once it's a great sprawling Democracy.
AI government turns in anarchy: The AI gets no bonus for the lower three difficulty levels, and has to stay in anarchy just as long as a player would. At the higher difficulty levels it gets a nice bonus; at Deity level the max turns an AI civ stays in anarchy is 2, almost as good as being Religious.
AI cost factor: This determines the cost factor for AI growth, shields and research. The cost factor for players is always 10; the lowest difficulty levels give a bonus to the player, while the top three difficulty levels give a decided bonus to the AI.
AI to AI trade rate: The AI thinks it is getting "better deals" from other AI civs than it would from you, on every difficulty level. If you're playing a Regent game and one AI civ wants something from another AI civ that would normally cost 130, the first civ only has to be able to pay 100 to get it. In the words of the editor, "This rate directly affects the frequency and aggressiveness with which AI civs trade amongst themselves."
AI and player factors: These aren't factors where the AI can get a bonus, but they're on the same screen and deal with difficulty levels so I included them. It's all pretty basic stuff, although the last one is interesting, which determines how many cities you can make before you hit the higher corruption level. Higher difficulty levels lower this number, which is based on the optimal number of cities per world size which is also listed. In short, you have to make do with fewer cities on the higher difficulty levels unless you want to deal with more corruption. This is true for both the AI civs and for players; the number of cities that can be built before corruption kicks in is also raised (by 25% post-1.29, I believe) for civs which are Commercial.
The AI bonuses are quite strong at the higher difficulty levels, apart from the other ones mentioned elsewhere (knowing where all the units are, knowing where all current and future resources will be, etc.) The fun, of course, comes in beating it even with all these bonuses. Perhaps in the next rendition (or more likely by Civ X) we can play against an AI that is adaptive enough it won't need all these bonuses to beat humans.
Gold: The tile must not have any type of resource or luxury on it.
Maps: always available
Nothing: always available
Settler: Player must not have a settler (active or in production) or any unit with the Settle AI strategy.
Player must have less cities than (TotalCities / NumActivePlayers).
Mercenaries (skilled warrior): There must be a unit available to the Barbarians as well as the player and that unit must be able to be built (or have been built) by some player in the game.
Tech: Player must still be in Ancient Times.
Barbarians: Player must not have Expansionist trait. There must not be a city within a 1-tile radius. The player must have at least 1 city. The player must have at least 1 military unit. The unit popping the hut must not have the "All Terrain As Roads" ability.