A long time ago I wrote the following wiki page, following a variety of conversations with Satoshi on the topic:

https://en.bitcoin.it/wiki/Contracts

Quoting from the top:

A distributed contract is a method of using Bitcoin to form agreements with people via the block chain. Contracts don't make anything possible that was previously impossible, but rather, they allow you to solve common problems in a way that minimizes trust. Minimal trust often makes things more convenient by allowing human judgements to be taken out of the loop, thus allowing complete automation.

Let's take one example, 2-of-3 dispute mediation. Elsewhere in this thread, some people are saying things like "if you don't trust your escrow agent with your money, why do you trust them to make a decision". These are very different kinds of trust. Typically online payment networks bundle them together. You trust your bank and credit card company with your money (more accurately, you don't trust them, but you know ultimately the costs of their failures will be socialised), and then because they are always the middleman you end up trusting them to do dispute mediation as well.

The problem is, they often aren't very good at it. Companies like PayPal tend to see dispute mediation as a cost of business rather than their core competency. Innovation in this space is rare. PayPal's policies, for example, make selling something to someone else who is standing in front of you impossible - you must be able to prove you mailed the item and if you can't, you automatically lose the dispute and the payment will be reversed. This policy is so deeply unintuitive that there's a whole industry of scammers out there who steal things from people by exploiting it. There isn't much scope here for doing anything better: payment networks tend to have a small set of one-size-fits-all policies and you can't go to the market to find a new one.

Now consider the Bitcoin approach. For many transfers there is no possibility of dispute mediation and thus none of the overheads associated with it. This often makes sense: either you know there will be no dispute (e.g. a gift from a family member) or you have other ways to resolve disputes (e.g. salary payments). When you want dispute mediation however you can agree on a mutually satisfactory mediator.

This mediator has very limited power. Contrary to what other people here seem to think, this does matter. For example, the mediator cannot steal the money, thus they make a poor target for hackers. If the mediator is a company there is very little temptation for bad insiders. They are not accepting deposits so all the complicated regulations that are triggered by that act don't apply. There is no possibility of them lending the money in escrow out and going fractional reserve.

Really all they can do is resolve a dispute badly, but there's no profit in that. So by preventing the mediator from accessing the money, all kinds of possible failure modes are avoided. In turn this means you can access a far wider set of possible mediators. For example, if you are trying to buy an antique violin over the internet PayPal's mediation policies are less than ideal:

http://www.theguardian.com/world/2012/jan/04/paypal-buyer-de...

With Bitcoin you could ask an actual expert in antique violins to act as the mediator. They do not need to be a computer security expert, they don't need to worry about becoming a target for physical theft or extortion, they don't need to worry they might accidentally lose the keys and destroy the money. There's tons of things they don't need to worry about. They could (if it existed) just use a friendly desktop app to pick a winner in case of a dispute. That means they can spend all their time inventing precise and well designed ways to resolve any dispute, fully leveraging their domain knowledge.

That's just one example of how redistributing and carefully controlling trust relationships can result in better outcomes.