The Settlers of Catan was one of the first modern European-style board games to become a worldwide success, in no small part due to the social aspect of the trading that takes place in the game. While many modern Eurogames have production of resources and/or the use of resources to build things, Settlers of Catan has a large focus on trading.
Here are some Settlers strategies on how to take advantage of trading:
Trade a lot.
This is the most important thing about trading in Settlers -- perhaps even more important than squeezing an advantage over the other player out of every trade. Even if you never gain advantage over your trade partner and all trades are perfectly fair, whichever player makes the most (mutually beneficial) trades will win.
To put it simply, imagine every trade you made gave each player in the trade a point. Even though these trades would be perfectly fair, whoever traded this way the most would get the most points and win the game. While fair trading will not always give you an immediate point, the basic idea still holds.
Claim advantage in trades.
Naturally, you shouldn't trade unless it benefits you somewhat. But if your opponent needs to trade because they are looking for a particular resource, this puts you in a position of bargaining power. When you hold the power in a negotiation, you should bend it to your advantage. Sometimes this can be as simple as asking your opponent to throw in an extra card of their choice to a trade. However, there are other ways to claim advantage in a trade as well.
Trade for the resources that are worth more.
Yes, your first priority is to get whatever resources you need for what you are trying to build. But beyond that, you should be trading to get better resources.
What are better resources? The ones that everyone wants and nobody has. Late in the game, this will be wheat and ore. Early in the game, it might tend towards wood and brick. Sheep will generally be less valuable than other resources.
But this will also depend on the distribution of numbers -- if a resource is on a six and an eight, it will be more plentiful and worth less. If a resource is only available on fours and twelves, it will be hard to get and quickly become valuable.
Trade on your turn when possible.
Trading on your turn gives you an automatic advantage because you can use the resources you gain immediately to build more roads, settlements, development cards, or what have you. In addition, by trading for what you need and then using it all on your turn, you won't leave yourself open to the ravages of the robber before you can use the resources you traded for.
Avoid the robber by trading.
Generally speaking, more resources is better, but if you have eight cards in hand when the robber is rolled, you'll wish you only had six. If your turn is next and you have the right resources to build what you want to build, it might be worth trading your extra cards 2:1 with another player to assure that a robber doesn't cut your hand in half.
Trade at 2:1 Ports when possible.
Rather than making a 2:1 trades with other players, trading 2:1 at a port will still give you the resource you need, without giving any of your opponents what they need.
All else being equal, it's better to trade with the bank than another player. In order to trade with another player instead of a port, that player needs to offer you something significantly better. (e.g., at least a 1:1 trade).
Blockade trading with the person about to win.
This goes double if that player is the active player. If Bob has 9 victory points, an un-upgraded settlement, and is asking for ore, do not trade with him. It doesn't matter if he trades you five cards for a single ore, because unless you can use your five cards to win the game immediately as the active player, chances are good that Bob will use your ore to win.
Blockade trades with other players strategically.
As mentioned earlier, generally speaking you want to do lots of trading. But just like selling weapons to a country that will use them against you is a mistake, you don't want to trade brick and wood to someone who will then build a road to cut you off.
Especially early on, don't hand brick and wood to players building roads towards the same areas as you. Later in the game, if you have managed to gain the longest road or largest army, don't trade your closest competitor enough resources to surpass you and steal your victory points.