I recently taught a couple of old friends Dominion, and it wasn't perfect, but we played a game in <45min, and all had scores from 20-30, so they obviously got the idea. Obviously many people are able to pick up a game like this with just a rules explanation and couple of rounds of examples, but I think it's still useful to consider what things make it easier to understand, both so people who are less predisposed to learn, but might enjoy it very much, can do so, and so if you're explaining something much more complicated, you can simplify anything not necessary in order to concentrate on the hard stuff.
I think many of the same lessons may apply to (good or bad) complex games like Bridge or Magic:TG.
1. "You start with a deck of ten cards, and the aim is to build it into a bigger, better deck. You add victory cards which don't do anything in the game, but the player who has the most at the end wins. You have money, that you use to buy victory cards and other things. And you have action cards, that do something specific, I'll explain in a second."
Keep it simple. Most people are overwhelmed by too much information, so give the information in order of importance. At this point, there's almost no benefit to considering whether cards can be BOTH victory AND action cards, or knowing the strategy of what sorts of cards to buy, but start with giving the idea that "you want more cards in your deck" and work up from there.
2. "In your turn, you have five cards from your deck, and you get to use them to buy more cards, so the aim is to have your deck full of as many useful cards as possible. You possibly play an action card, and then buy a new card costing up to the amount of money you have in your hand. Afterwards all five cards go in the discards, which will eventually be shuffled into your deck again."
Almost everyone benefits from being given the general idea. They may or may not benefit from being given a precise (if longwinded) statement of the exact rules as well.
"Play an action card, and then buy a new card" is not true, but it's a useful description of a basic, typical turn, and most people understand things much better as differences from the norm.
The two things most people are confused about is (1) if you use two cards, do you keep the rest in hand? and (2) when you spend money, does it go away permanently. Don't try to critique these ideas, as it'll just add more confusion, but make sure the basic explanation contains the basic idea that the cards always go round and round.
3. "Action cards do things like let you draw extra cards from your deck into your hand, or give you extra money for that turn.
And then explain one or two action cards a turn as people first have the money to be able to buy them.
4. "OK, you probably want to buy [silver] or [village] but you could also buy [one of these] if you want
People are paralysed by too much choice, so giving a choice between the probable best option, and indicating there are other options there if they're interested, is often MORE helpful.
Likewise for the whole thing, say "the normal aim of the game is to get enough money to buy gold, and then enough gold to buy provinces". You can do other things, but that's a good framework to start thinking from.
5. Lay out 5 simple-ish kingdom cards
If you've never played a game LIKE this before, 5 kingdom cards is normally enough, even if that doesn't give you a lot of choice. Leave out anything that people are often confused about, even if it's cool. As a rule-of-thumb,
6. If people put their played cards straight into the discards, or shuffle the discards into a too-small deck, then let it go.
This is a difficult call, because if they're following what's going on, you don't want them to start extrapolating from wrong premises. But on the other hand, if they trust you, they're likely to be happy to follow your directions on things they don't have any general conclusions about, so if you just say "shuffle" or whatever, it's fine, and if they don't follow the rules that only matter very occasionally, then explaining the difference just adds more information that they (may be) not ready to assimilate.
7. Don't play too hard.
Do obvious things yourself. Buy silver and gold. Buy several copies of simple action cards. By all means win. But try to avoid long chains of action cards or anything else if you think the other people won't follow it.
Obviously all this isn't always necessary, but I think it's helpful to have this sort of advice in mind for when you need it.
I think many of the same lessons may apply to (good or bad) complex games like Bridge or Magic:TG.
1. "You start with a deck of ten cards, and the aim is to build it into a bigger, better deck. You add victory cards which don't do anything in the game, but the player who has the most at the end wins. You have money, that you use to buy victory cards and other things. And you have action cards, that do something specific, I'll explain in a second."
Keep it simple. Most people are overwhelmed by too much information, so give the information in order of importance. At this point, there's almost no benefit to considering whether cards can be BOTH victory AND action cards, or knowing the strategy of what sorts of cards to buy, but start with giving the idea that "you want more cards in your deck" and work up from there.
2. "In your turn, you have five cards from your deck, and you get to use them to buy more cards, so the aim is to have your deck full of as many useful cards as possible. You possibly play an action card, and then buy a new card costing up to the amount of money you have in your hand. Afterwards all five cards go in the discards, which will eventually be shuffled into your deck again."
Almost everyone benefits from being given the general idea. They may or may not benefit from being given a precise (if longwinded) statement of the exact rules as well.
"Play an action card, and then buy a new card" is not true, but it's a useful description of a basic, typical turn, and most people understand things much better as differences from the norm.
The two things most people are confused about is (1) if you use two cards, do you keep the rest in hand? and (2) when you spend money, does it go away permanently. Don't try to critique these ideas, as it'll just add more confusion, but make sure the basic explanation contains the basic idea that the cards always go round and round.
3. "Action cards do things like let you draw extra cards from your deck into your hand, or give you extra money for that turn.
And then explain one or two action cards a turn as people first have the money to be able to buy them.
4. "OK, you probably want to buy [silver] or [village] but you could also buy [one of these] if you want
People are paralysed by too much choice, so giving a choice between the probable best option, and indicating there are other options there if they're interested, is often MORE helpful.
Likewise for the whole thing, say "the normal aim of the game is to get enough money to buy gold, and then enough gold to buy provinces". You can do other things, but that's a good framework to start thinking from.
5. Lay out 5 simple-ish kingdom cards
If you've never played a game LIKE this before, 5 kingdom cards is normally enough, even if that doesn't give you a lot of choice. Leave out anything that people are often confused about, even if it's cool. As a rule-of-thumb,
6. If people put their played cards straight into the discards, or shuffle the discards into a too-small deck, then let it go.
This is a difficult call, because if they're following what's going on, you don't want them to start extrapolating from wrong premises. But on the other hand, if they trust you, they're likely to be happy to follow your directions on things they don't have any general conclusions about, so if you just say "shuffle" or whatever, it's fine, and if they don't follow the rules that only matter very occasionally, then explaining the difference just adds more information that they (may be) not ready to assimilate.
7. Don't play too hard.
Do obvious things yourself. Buy silver and gold. Buy several copies of simple action cards. By all means win. But try to avoid long chains of action cards or anything else if you think the other people won't follow it.
Obviously all this isn't always necessary, but I think it's helpful to have this sort of advice in mind for when you need it.