The first thing people need to realise is that 'the economy' is not a monolithic thing. The very word itself can be quite misleading - most people associate 'economy' with low-prices, or money-saving. And some people have a vague notion of a cloud-like thing called 'national economy'.
If you were truly self-sufficient; you grew, raised and hunted your own food, you made your own buildings, you designed and manufactured your own furniture, cutlery, utensils, what you wore , etc etc etc, you had your own oil well, doctored your own cuts... well, if you could really live like that, then you won't need to be part of any exchange or trade transaction. But because this is not the case for the vast majority of people, we tend to stick to doing what we're best at doing, and trading with someone else (exchanging of goods or services) for what we need. These transactions, taken as a whole across a particular area, is what constitutes that area's economy.
Let's look at the process of exchange. The fundamental exchange is one of barter; I have chickens which you want, you have goats which I want. We negotiate and haggle for a bit, but eventually we agree that I will give you 10 chickens in exchange for 2 goats. Putting aside such complications as perhaps I would have been willing to trade 12 chickens for 2 goats or that if I threw in another chicken you would have given me 3 goats (price elasticity of demand/supply), this is an economic transaction, and again, the economy is made of all of these transactions taking place over a given location.
Believe it or not, our current economy can still be simplified into barter exchanges. The only difference is, we have all more or less agreed that by and large, ONE side of these exchanges will involve a common commodity - money. When a person is exchanging money for something else, we call him the buyer. The one receiving the money, we call him the seller. Only, this hides the reality that we are both buying and selling - I am selling my money for a product, and the other dude is buying my money with his product.
The thing that happens with a money economy is this - because money is now the fundamental and common form of value exchange, the size and health of the economy can be tracked through the amount and velocity of money. We'll get back to that.
Banks are strange companies. They buy and sell money, much like moneychangers, but instead of giving you a service in return, they buy and sell money with... more money, or the promise thereof. You see, when you deposit money into a savings account, or a fixed/term deposit, you're actually selling your money to the bank for 8% (or whatever rate your bank offers) more of your money back in a year's time. They go out and sell that money for even more money, which is what we call loans. But where does that money eventually end up? Right, either in that very same bank, or a different one. It is exceedingly strange, isn't it? Wealth and money appear from nowhere, seemingly. You can even calculate the theoretical amount of 'come from nowhere' money generated in this way - it's called the money multiplier.
Or rather, it appears from the future. And the perception of the future. And regardless of whatever had happened to cause people's perception of the future to change (i.e. there isn't enough money in the future), the credit crisis means this;
1. The amount of money decreases as people withdraw their funds from the banks; once because actual currency is being withdrawn from circulation, and a fair bit more due to the loss of the multiplier effect. This shrinks the economy.
2. The velocity of money reduces, as people start hoarding their money; this reduces the number of transactions out in the marketplace, and shrinks the economy.
Is the bailout needed? My gut feeling says yes, but it is more the idea of the bailout that is needed. If you can restore confidence in the future, the problem that is to come won't be so bad. Again, we have made money the common form of value exchange. We need it to continue circulating, and in similar quantities as before.