Owing money to a creditor is a serious matter. It's important to realize that there are two types of debt collection companies, debt collection agencies and debt collection law firms.
Borgo describes agencies well, so I'll pick up with what happens when you don't pay an agency (or original creditor) to whom you owe money. If the debt is small (under $500 is the rough cutoff here in the midwest) then you just get some nasty words on your credit report. Debts that small are almost invariably from high interest check cashing joints.
Any bigger than $500, and the agency will probably pass your debt on to a law firm. As an IT employee of such a law firm, I think I can provide a somewhat informative picture of what these firms are like, and what is involved in being sued by one.
Disclaimer: I don't guarantee any of my above or below statements to be true. I do not directly work in debt collection, but it goes on all around me every day.

What a Debt Collection Firm is

The firm isn't a big cubicle farm of low wage saps reading from scripts. In fact, we don't call you, you call us. The firm is composed entirely of lawyers, paralegals, and support staff. You will speak almost exclusively to paralegals, should you call. Debt collection is profitable in good times and rediculously profitable in bad times, so the people at every level will always be well paid, and usually very good at what they do.

From Debt to Lawsuit

When the firm gets your debt, a quick and dirty check is run on you. The firm will find out if you own real estate, if you work, if you get social security or disability, how much money you owe to other companies, and whether or not you are paying those companies. If the firm decides there's a realistic chance of recovering the debt from you (and there is, in almost every case) then a guy in the firm will say "OK to proceed with suit" and the fun begins.

A Quick Guide to the Debt Lawsuit Process

Skipping the technical details to just speak in general terms, the firm will file suit in your county's court system and you will be served a summons by certified mail which informs you that you are being sued. You have about three weeks to respond to the suit. Most often this involves you agreeing to pay the firm either in full or in monthly installments. You can also fight the lawsuit. Fighting the lawsuit means you have hired a lawyer, who will be much more qualified than me to explain things, so I'll skip that part.

What happens if you don't reply to the Lawsuit

If you make no formal response to the lawsuit, the firm will get a default judgement against you, where basically the court recognizes that you owe the firm the full ammount of money in question. The interest has been ticking, and attorney fees are added. Then the firm will use a few different techniques to get you to pay:
Wage Garnishment: If you are employed, the very first thing the firm will do is get a wage garnishment order against you. Your employer will be legally required to deduct an ammount of money (about 20% of your after taxes earnings) from your paycheck. This hurts.
Bank Garnishment: If you have money in the bank, the firm will get that.
Real Estate Lien: If you own real estate, the firm will put a lien on it, where the court gives them the right to collect the debt through your property.
Executions: If you own a business, the firm will have the sheriff's office come in and take the money out of your cash register.

Driver's Liscense Revocation: Yes, you can have your liscense suspended for having an unpaid civil judgement! Public disclosuer: We don't do this, but many firms will give notice in your local paper that they are suing you. In small towns the papers often will report these lawsuits even if they aren't asked to do so. My Advice

Pay your debt! Despite what various 'get out of debt lawsuit' books will claim, if you can pay, we will probably make you pay. The definition of cannot pay is: "Debtor is unemployed, lives on an income that legally cannot be garnished (social security, welfare or disability), debtor owns no real estate and no valuable property". So you can get off without paying, but what's the point of going in debt in the first place if you're going to live with almost nothing?
At any rate, here is how I would suggest you minimize the amount you end up paying:
* Pay early! Interest and attorney fees are mounting every day you wait. Every resource the firm uses to make you pay will be added - at a high cost - to your bill. Putting off paying or even communicating with the firm in that hope doing so will get you off the hook is the dumbest thing you can do. You'll pay anyway, and owe twice as much when you do. Often your best bet will just be to get a decent loan from the bank and pay off your creditors. Legal interest is usually 10% and often much more, the bank can usually give you a better deal.
* Settle in full. Firms like it when you do that, and will usually cut you a better deal.
* Negotiate! Even if you are a poor debater, the paralegal you call to discuss payment will be very willing to lower the amount you have to pay to get a settlement. If you settle in full, you can often expect to pay just 80-90% of the amount owed. Many creditors will accept 80%, but it doesn't hurt to plead financial difficulty and make a lower offer (in which case the person will usually have to get approval from his boss).
* In cases where that amount is ridiculously inflated by high interest and attorney fees, I've seen people talk their way down to just paying 20-30% of the on paper amount owed.
* If you do reply formally to the lawsuit, you must send the letter to court, and actually deny the allegations made against you, not just explain why you aren't going to pay, otherwise the lawyer can more or less ignore your letter.
* It many cases, bankruptcy is going to be your best choice. It's messy but it often is better than having your wages garnished for a decade or two. However, I think bankruptcy is only a legitimate thing to do if you are a victim of predatory lending, going through a messy divorce, injured and unable to work, or experiencing some similar sort of hardship.
* If you owe a lot, get a lawyer. A debt collection firm handles tens of thousands of suits a year, and will occasionally screw up. They will always fix the mistake sooner or later, but a good lawyer can quickly spot that error and pounce on it, and often get the case dismissed before the firm can react.