Note: This writeup focuses on general insurance, which is in contrast to life insurance, which may differ. Life insurance covers you when something happens (ie your dying or surviving to a certain age) as opposed to general insurance which covers you if something happens. The principle source of this information is my experience from working in the motor insurance industry in the UK, so the examples I use are from that side of things. Most insurances throughout the world are based on the same models and are subject to comparable laws. IANAL YMMV, etc etc.

A writeup above, and the general feeling of many is that insurance is like a kind of bet or wager that you have to take, and you will probably lose. Whilst it certainly seems that way, and it can humoursly (or cynically) looked at in that manner, there is a fundamental difference between the two.

There is, believe it or not, a set of principles of insurance. The difference between a wager and insurance is the principle of insurable interest.

Insurable interest

This principle basically defines what can and cannot be insured by any party. Basically, the party seeking to insure something must have a financial interest in the thing being insured. If the object being insured is damaged, destroyed or stolen etc the insured must suffer financially unless insurance was taken out.

What that boils down to is that I can't insure your vehicle from getting in a crash. If I did, that would be a wager "I bet you'll crash your vehicle". I would gain if the vehicle was damaged, and would not lose if I wasn't insured.

Insurable interest is not always obvious. You don't have to own something to have an insurable interest in it. You might have borrowed something which you are now responsible for the wellbeing of. If the object is damaged or stolen, you would be expected by the owner to replace or repair the item - thus you would have insurable interest. Likewise, a repairer holding a customer's property for repairs has an insurable interest in the property (for example a garage repairing someone's vehicle) since they would be held liable should anything happen to it.

Indemnity

The purpose of insurance is to put the insured back to the same financial situation they were in immediately before the loss. This may seem obvious but it has implications. First off it means that if you insure something twice, you don't get paid twice. The two insurance companies may both put some money towards the payout (this is covered later). The idea here is that one should not profit from insurance.

The issue of excess (or deductible as it also known) should be raised here. The excess represents an agreement the insured takes out to be held liable for part of any loss that arises. With an excess of £200 the insured has agreed to pay the first £200 of any claim. This helps keeps premiums low, and prevents lots and lots of little claims being made, driving premiums even higher.

If a claim is made, and the insurer discovers the insured was underinsured then the difference might be deducted from any settlement. An example from car insurance:- You buy a car for your 18 year old son, and then you become a policyholder for the vehicle and name your son as an additional driver (even though he is the only or main driver of the vehicle). This is known as a 'fronted policy', and is technically fraud. If you are discovered you may find that the insurance won't pay out. They may still settle with you but they would subtract from the settlement the difference in premiums had your son been declared as the main driver. I don't recommend this - it will undoubtedly be recorded and may affect future insurances...and you can bet any future deviations of good faith will not be handled so amicably.

Good Faith

Good faith is basically another way of saying "You promise not to commit insurance fraud". OK, it goes deeper than that. First, the insurance company should act in good faith. Attempting to resolve issues quickly, process claims in a timely manner, and look for reasons to pay out a claim rather than reasons for denying it. Second, the insured should not omit material facts that could affect the risk the insurer is accepting. Previous claims, job descriptions, usage of the vehicle (how many miles its expected to do, if its for business use etc) etc.

It's not just omission but commission that can be a breach of utmost good faith. This would take the form of actively trying to commit fraud.

Subrogation

This is the most misunderstood concept of insurance in my experience. If someone is rear ended at a set of traffic lights, they might expect their insurance company to claim from the other party's insurance on their behalf. This is not strictly speaking true because of subrogation.

If you claim from your own insurance company and received full indemnity then the loss becomes the insurance company's, not yours. You have no longer suffered a loss, so you cannot pursue any third parties. Your insurance company is now the one who has suffered the loss - they have subrogated (traded places so to speak) with you. The insurance company, therefore, has the right to recover that loss and they can not pursue it should they so choose.

This leads to some upset people who got the wrong end of the stick. In cases where the insured does not feel the incident was their fault, the liability might get split and they have a claim on their policy. This is an important point: Your insurance company is not an accident management company. All they have to do is settle the claim with you, they are not obligated to manage everything for you (unless specified in the documentation) and they can choose to settle for less than their costs if they choose. If you allow your insurance company to subrogate the loss, you are no longer in control of the outcome of the claim. This is an important point that needs bolding and italicizing, it really is that important.

If you want to avoid the risk of a claim being made on your policy for a non-fault incident, don't have your own insurance indemnify you. You are perfectly entitled to claim your costs from the third party insurers yourself (or use an accident management company to do it for you).

One further point, that I should mention here. There are occasions where insurance companies have not pursued a third party with suitable vigour, and the financial ombudsman succesfully (for the insured) intervened. Don't count on this though.

Contribution

I discussed contribution partially in the indemnity section. If something is covered twice or more the insurers will divide the value of the claim between them and both contribute towards it. For example, if you have a jacket stolen from your car, you might be able to claim on both the car insurance and the home insurance. If the jacket was worth £100, the two insurance companies may contribute £50 each.

Insurance companies rip us off

I'm sure it may seem like that, but its not necessarily true. In most (if not all) countries the premiums and the costs of the claims should roughly balance. Insurance companies cannot make a profit by selling insurance. Crazy, huh? To illustrate this we turn to the UK motor insurance industry. In one recent year motor insurers paid out £100,000,000 more than they took in premiums. If we look at the trends over the past few years we see this is not uncommon.
Year              Underwriting result
1992                    Loss
1993                    Gain
1994                    Gain
1995                Slight Loss
1996                    Loss
1997                    Loss
1998                    Loss
1999                    Loss 
2000                    Loss
2001                    Loss
2002                Slight Loss
I like to think of it as the baker's dozen of insurance. Insurance companies can't be seen to be taking more than they pay out - otherwise they are making a profit from insurance. A phenomenon known as the insurance cycle comes into play here as well. Insurance companies reducing premiums to the point where they are making a loss just to increase their market share. Like banks, the majority of profits come from stock trading with the money they have access to rather than by selling something for more than it is worth. Indeed, in the UK insurance investment accounts for 17% of all investments in the stock market.

Rising Premiums

Premiums are almost always rising. Why is that? Well it depends, in the motor trade the two main factors for this are personal injury claims which are on the increase in many countries with the 'no win, no fee' explosion. This has been popular in the States for some time, so I'm not sure whether injury claims are on the increase there or not. Another factor is the price of parts which also generally rises.

Tying in with personal injury claims of course is the ever present shadow of fraud. Some studies show that 7% of people have committed insurance fraud and most people would commit insurance fraud if the opportunity presented itself.

Finally, there is the problem of uninsured drivers. There is little that can be done about these leeches, but here in the UK the Motor Insurers Database helps the police quickly check if a vehicle or driver is insured. The Motor Insurers Bureau provides a means to get indemnified for a non fault incident involving an untraced or uninsured driver without having to make a claim on your own insurance.

Sources:
Most of the information comes from my training and experience in insurance. As such it might be inaccurate. Please inform me of any mistakes, or if your country does things radically different. Statistics and figures came from http://www.abi.org.uk