The original rules of Settlers of Catan are fine as they stand. However, there are always tweaks you can add to make the game better suit your playing style. You can find a few of these below. Thanks so much to everyone for contributing the variants you like to play by. I'd love to hear more, just post your suggestions on the forum.
Too much robber will stunt your growth - I recommend this variant to all new players. If you play with it, then until every player has taken their turn twice, every time a seven is rolled, the player who rolled it must immediately re-roll and continue to do so until they roll something other than a seven. While the robber is nice for stopping players from taking a large lead, it is no fun for anyone in the early game. It always leaves a bad taste in my mouth when the seven is rolled early on because not only does it mean no new resources are generated, but, by placing it on a tile, you have decreased the overall production of the board. With this variant, most players should have at least come close to expanding once by the time the robber comes to ruin their fun. Alternatively, you can choose to ignore the first three or four times the seven is rolled. Or, you could re-roll all sevens until every player has 3 VP (or one player has reached 4). If you want the game to startup even quicker, consider letting players gain resources for both of the initial settlements played, instead of just the second.
Home is where the robber is - This variation on the rules allows a player to choose to keep the robber where he is after rolling a seven, instead of being forced to move him. The player then selects a resource card from one of his/her opponents on the tile the robber is at, just as if he/she had just moved the robber there. This alternate rule increases the power of the robber and makes him better at stopping those pesky runaway leaders.
With Many VP Comes Much Responsibility - This rule states that when taking resources from a player after the robber was placed on him/her, the amount you take is based on the difference between the number of victory points the person placing the robber has and the number of VP's the person being robbed has. If the difference is 1 or less, then take one resource as normal. A difference of 2-3 means the robber steals 2 resources (both still randomly selected). Finally, a difference of 4+ means the robber takes a whopping three resources. This rule is another one which empowers the robber, this time encouraging the players far behind to nail the leaders. It's fun to play with, though some people don't like to have to deal with the extra time it takes to figure out how many resources to steal.
Bribery - Considered immoral by some, a way of life by others. This bit of corruption weaves its way into Catan with the alternate rule allowing players to give resources to one another in exchange for services. Normally you cannot bribe for development cards, though that's an extension of the alternate rule which you could play by. Normally you would bribe another player to not put the robber on you, or to break an opponent's road in half, or even to not compete with you on a spot where you're trying to build a settlement. You could even allow players to offer specific resources when you place the robber on them instead of you picking a random card from their hand.
Share Ports - If you play with this rule, then when it is your turn you can allow other players to use any ports you have access to. You may force them to pay you resources to do so however.
Don't Waste - Recycle! - This alternate rule is for those of you who cringe whenever you see resources being discarded back into the bank when a seven is rolled. Instead of throwing them out, the players still pick half of their cards to discard, but give them to the player who rolled the seven. However, if the person who rolled the seven had to discard cards as well, then everything goes to the bank. This alternate rule can give a quick boost to a player who is having a rough time getting resources, allowing them to get back into the game.
Explore the Wilderness - When you lay out the board at the beginning of the game, put all tiles face down. Then, place the numbered tokens down as normal, skipping a number of tiles equal to the number of deserts in the game. Whenever a settlement or city is placed on an intersection, reveal all upside-down tiles it touches. If one is a desert, switch it with one of the tiles that has no number token on it. This variant makes expanding much more fun, though it adds a lot of luck to the game. To switch things up even more you could have the deserts act as gold mines and count any tiles that a token wasn't placed on as deserts (don't flip them over when a settlement/city is built next to them)
Linked 12's and 2's - In this variant whenever a 12 or 2 is rolled, act as if both were rolled and give everyone resources who has a settlement/city on tiles with either of those numbers.
Randomized Abundance - This is another exploration-based variant. After you have placed all the tiles in the game, randomly shuffle up the numbered tokens and place them face down on all the tiles except deserts. Once all players have put down their first settlement, flip over all tokens they are touching. Then, go through the second round of settling and flip over any new tokens touching a settlement/city. For the rest of the game, flip over a token as soon as a settlement/city is created next to it. This variant involves a heavy amount of chance, though, as you could start next to two red tokens, or next to two 12's.
Can't we all just get along? - These are a couple of variants on using the robber which encourage cooperation over competition. First, when you roll a seven, if you put the robber on a tile with no settlements/cities next to it then you can take a resource of the same type as that tile. The second option is to remove the robber from the game entirely. Instead, when you roll a seven, you get one resource of whatever type you want from the bank. These variants can be helpful if you want to make the game more light and fun, but will allow for someone who takes an early lead to win much more easily.
Banking - If you are sick of losing extra resources when a seven is rolled then try this variant. On your turn you can pay another player some resources for them to take any number of resources from your hand into his/hers and not to look at or use them. However, they still count towards his/her hand size. Then, when it is your turn again, the other player must return the cards to you.
Port Block - In this variation any ports adjacent to the tile that the robber is on are unable to be used by any player. However, players may use other ports which they have access to, and can always do a 4:1 trade. To make the robber even more powerful, you can also play with the rule that no settlements, cities, or roads may be built which touch the hex that the robber is currently on.