Here are the two main approaches:
First time penalty abatement is the low hanging fruit of penalty waivers. If you are compliant on your taxes, filing and paying recent years, and also have a solid history before you fell behind on taxes, you and/or your tax professional should be able to waive penalties for one year. This can be done in a phone call to the IRS or thru correspondence. But you want make sure you qualify so you don't waste your time. IRS Transcript subscription services like the one WATAX uses lets us know exactly whether you qualify and for what year.
Waiver of failure to pay, failure to file or accuracy related penalty abatements are quite harder. If you are hoping to waive penalties on multiple years, you will need to describe and prove "reasonable cause" why you fell behind, other extenuating circumstances explained thoroughly and also prove current tax compliance. Any request needs to be sent to the IRS service center where you mail your tax returns or thru the IRS agent assigned to your case. All requests need to be signed "under the penalty of perjury, I can attest that all of the facts contained in this statement are true...." If your original request is denied and you feel you had a strong story and reasonable cause, you should definitely APPEAL your rejected penalty waiver! Sometimes its good to get a live body/heart/mind from the IRS to understand your plight. But don't waste the IRS's time in appeals if you don't meet the criteria.