What are some rule changes that shorten the game, and what drawbacks do those changes have?
Please limit your answers to rules changes with which you have first-hand experience!
We've tried to replace the robber with a more benevolent figure (we used a LEGO Dumbledore microfig but you can just reuse the robber figure and call it, say, a magician)
We came up with that because we don't like the robber that much, and it sped up the game quite a bit. Now, I suppose any house rule which gives more ressources would shorten the game length, but with that one is slightly strategic in nature.
Of course there's also the bog-standard way to change the duration of the game: instead of 10 victory points, aim for less (8 for example). As long as you agree before starting the game :-)
We usually play Settlers of Catan with resource cards exposed. This dramatically shortens the trading, as there is no question of whether any player owns a particular resource. The 'drawback' of this, is that players no longer have to remember what's been received.
Another tactic we use to shorten the game is playing quickly. This isn't really a rules change, but if players agree at the beginning that they are hoping for a quick game, then this seems to give enough motivation for everyone to do so.
In the setup phase, place a settlement first, and then a city second. This is one of the common changes that many of the longer adaptations feature and it just helps you get going quicker at the slower/earlier stages of the game.
When we only have time for a short game, we will eliminate the robber and allow each play to take a resource of their choice on roll of 7. It cuts the time a lot, and eliminates a lot of frustration when playing with people who are inexperienced with the game. (I like the using a city for the second settlement idea, and the gold if no resources. I want to try them!)
My preferred way to play is having 2 resource productions before each player's turn, which will come close to halving the length of the game. This takes out most of the waiting, because you will almost always be building something new every single turn.
The only problem is that its harder to get rid of your resources before a robber hits as you have to 'survive' twice as many potential times where a 7 might come up and make you discard. To balance this, you could allow an 'intermediary building phase' at the end of every player's turn where all players are allowed to build, but are only allowed to trade with the active player (this is how the 5-6 player expansion works).
If you really want to make the game quicker, you can eliminate the concept of turns altogether and let everyone build and trade at once after a designated number of resource productions happen, but this will make for a bit more chaos as players race to get the best trades and build a settlement in a choice spot as they no longer have to wait their turn before doing so. However, if your group likes a tenser, fast-paced game it's worth a shot.
I personally prefer to use a deck of cards with the same probabilities instead of rolling dice. Its a lot faster just to draw the top card of the deck, and its a ton more flexible because you can choose to go through the entire deck before reshuffling, so each number is guaranteed to come up a certain number of times. If your group doesn't having to randomly discard resources a lot, just remove a 7 or two from the deck. IMHO this adds a fair bit more strategy and flexibility that makes the game more enjoyable. :D
A rule variation that I prefer is to limit the movement of the robber:
When a 7 is rolled or an army played, can move the robber one hex position.
This speeds up the game with more resource production, since you can't automatically move the robber to the location where your opponents produce the most resources.
Drawback is that robber tends to stay in the same area of the board so one player can get stuck with the robber for much of the game. So this becomes another strategic factor in initial placement an village/city construction location.
Stolen from Settlers of America.
A new currency is added: Gold.
On any non-robber roll if you receive no resources you receive 1 gold. Gold can not be stolen, but it can be traded to other players (usually 1 gold to sweeten a deal), or traded (maximum of twice per turn) to the bank for any resource at rate of 2 gold=1 resource. This doesn't speed up the game a ton, but like most of the other answers putting more resources in the game results in a faster finish.
We've talked about (but have not play tested either of these yet) having a settlement on the desert give you 1 gold when the robber is rolled OR having any settlements on the desert gives you a single extra gold when you receive no resources (never more than 2 though, even if you have 3 cities around the desert.)
We have found that slowness has a varying impact. Slow is ok when it's not boring (unless dinner is burning while you're "nearly finished!" for an hour), but slow is awful when it's boring. The worst possible slow part is the very beginning, with endless fussing about which direction to build in, and the worst is those very first few moves. We therefore have a house rule:
This means that people still hum and haw about where to put the settlement, but they just plunk the road down immediately and we can move on. It also lowers the "I would have won except you cruelly blocked me on turn 2 and I had no chance I hate this game."
The other thing that slows a game tremendously is a very early robber release. When hardly any hexes are producing, stopping one has more impact than it does later on. So our second house rule:
This prevents that slowdown and also the associated "I would have won but you randomly chose me for that robber on turn 3 and I had no chance I hate this game."
Honestly, a game where one person is just stuck with two settlements, two or three road segments, and basically nothing for the whole game, while the others work their way up to 7, 8, 9 points before that person gets a 3rd or 4th point is not only slow, it's less fun. So avoiding the two biggest causes of someone cratering is a good thing. Some of our more serious-gamer friends mock us for these rules and say that's what happens when you let your children play, but we've noticed that they speed up games and make everyone happier, so we're keeping them. (And the friends are happy to take advantage of those rules when they play at our house.)
Simply change the goal.
You can basically do that with any "number of points to win" game, it's so simple no one ever thinks about that!
Play no-trade reverse robber.
No trade is straight forward enough. No trading with other players is allowed. This means all the negotiation in turns is removed, which means you're often just rolling, picking up your resources, and passing the dice. You are still allowed to trade with the bank/ports.
Reverse robber. When you roll a 7, you are not allowed to put the robber on anyone else, unless you forced to. ie. You must either place the robber on an empty tile, or place it on your own tile. If there are no empty tiles, or no tiles that aren't solely occupied by you, then you may put it on a tile that you share with someone, and steal from them.
There are two variants of this rule. RR is stricter and says whenever you move the robber, whether by 7 or by knight, you must place it on yourself/empty tile. RR7 says that you're allowed to place the robber where you want, when you're playing a knight.
The effect this rule has, is that the 6/8s tend not to be blocked, so resources flow more freely.
I have also heard of the playing with the resource cards face up but I hate this house rule because it takes away the fun of bluffing. Here's a couple suggestions:
The items above with
strikeout are ideas that I have not personally tried, at least not exactly as written. I didn't see that part of the question before. Feel free to edit/delete this answer if you like.