I owe around $20,000 to the IRS for underpaying income taxes for three years. I was self-employed and didn't understand how to figure out exactly what I owed. Is it possible to settle with the IRS, or am I going to have to pay the full amount?
The answer to your question is "yes—and maybe." That is, it is possible to settle with the IRS for less than the full amount; however, you may need to pay the entire debt.
In principal, settling with the IRS is like settling with any other creditor: you demonstrate to them that you cannot pay the full amount of the debt and offer them something less than that as payment in full. The IRS has a procedure for this, called an "Offer in Compromise."
The IRS does not have to be as flexible or accommodating as many other creditors, however. For example, tax debts are very hard to wipe out in bankruptcy, so the debtor lacks the leverage of being able to threaten to file bankruptcy. As a result, the IRS takes a somewhat hard line, and says that "an offer will not be accepted if the IRS believes that the liability can be paid in full as a lump sum or through a payment agreement." A payment agreement is basically a payment plan or schedule—it's an arrangement to pay the debt in installments, over time. If the IRS feels that this is a viable option for you, they are less likely to accept your offer in compromise.
Also, the IRS will not accept an offer in compromise if the offer is less than what they call the "reasonable collection potential," or RCP. The RCP is what they feel they could get if they used the legal process to garnish wages or force the sale of your assets. The net result is, while you can always try, you should only expect the IRS to accept your offer in compromise and settle your tax debt for less than the full amount if that really is the best they could expect to do.
You can enhance your chance of a successful outcome, however, by having experienced counsel negotiate for you. Settling with the IRS is an art; it requires an understanding of how agency representatives think, what their guidelines are, and what they have accepted as settlement previously. Someone who has done this before will be more successful than someone trying it for the first time.