Speaking from 10 years of experience in the car dealership/auto repair industry, some things to consider when purchasing extended warranties or service contracts (which I'll abbreviate as "ext. warranty" for convenience):
Like insurance, an ext. warranty is akin to making a bet. You are making a bet with the provider that something on your vehicle will fail that will be covered, and the provider is betting it will not. And just like a casino, the odds will always be tipped in the house's favor. Meaning, more often than not, people who purchase ext. warranties will spend more money on the warranty itself then they would on any potential repairs that would normally be covered.
To add to this, the profit margins on ext. warranties are HUGE. Even larger than the actual sale of the vehicle the ext. warranty is covering. This is why dealerships push them so hard. There are also huge incentives at both the employee and franchise level for dealerships to sell these, because of the large profit margins. Like eddiebrown found out, many dealers will jack up the price on these significantly, because any amount they sell the contracts for above the minimum sale price is direct profit that goes straight to the dealer.
Another important thing to keep in mind is that with ext. warranties is that you cede some amount of control over how your vehicle is repaired. This is especially true in the case of service contracts, which ARE NOT warranties, but closer to insurance companies in how they operate. Just like with regular vehicle insurance, providers will only cover repairs that they approve, and they will almost always repair the bare minimum needed to get the vehicle running again. For example, if your engine blew a head gasket, and the repair shop was recommending resurfacing the cylinder head, the service contract company may not approve the resurfacing (yes, this actually happened to a customer of mine). You also lose some ability to choose where you take your vehicle for repairs, though most decent service contract providers will work with just about any shop you take your vehicle to.
Generally speaking, the only times I have seen ext. warranties work out for people is when they are concerned that they cannot cover large repairs to their vehicle, and they need that extra layer of financial security. Many dealerships will add the cost of the ext. warranty to the vehicle loan, meaning a $2,000 coverage plan suddenly becomes more manageable when all it does is add a few extra bucks to your monthly payment. However, for those who have the financial means to cover larger repairs if they come up, it may make more sense for them to just keep that $2,000* in the bank. If repairs come up, you have the money. If you don't, you just saved yourself $2,000, plus any interest you made with that money in the bank.
Finally, BLUEKNIGHT911 touched on this, but it's important enough to reiterate: There is a significant difference between an extended warranty and a service contract. An extended warranty is offered by the manufacturer of the product you are getting the warranty for, and generally, they are easier to get coverage from. The manufacturer manages the claims process, and it's integrated with the dealerships, so generally the whole claim/repair process is smoother & faster. They also tend to only use OEM parts & repair techniques, which may be a plus or a minus depending on the situation or the preference of the customer. And, of course, extended warranties will generally only allow OEM dealerships to perform the repairs.
Service contract companies are provided & administered by a third party, which generally have little or nothing to do with the OEM. The biggest difference with these is that they will usually explicitly state in their contracts what they will cover. Some service contracts, called named exclusion service contracts, will list parts & repairs they will not cover. Be wary of the inexpensive service contract! Chances are, if you read the contact carefully, they only cover a limited number of components. A very common exclusion in service contracts is gaskets & seals (valve cover gaskets, o-rings, water pump gaskets, etc). The other thing service contract companies will tend to do more often is send out inspectors to verify a shop's diagnosis & quote. Depending on what the inspector reports to the service contract company, they can alter what is repaired or reject the claim entirely. On large repairs, such as blown head gaskets or repairs over a certain amount, most service contract companies will send an inspector automatically as a general policy.
Anyway, these have been my observations over the years. Just my .02, take them with a grain of salt, your mileage may vary, yada yada yada...
*Just an example. Amount you should save depends on how much you'd be willing to put into repairs before junking a vehicle, whether or not you use the vehicle for work/commuting or recreation, etc.