I've scoured the internet searching for an answer to this exact question. I'm not an expert, but it seems that a convicted felon CAN get a passport. Case in point, I know of a guy who after about a year after his release from prison travelled to Sierra Leone to visit his fiancee. He was still under probation during this time.
It seems that the passport is merely an international identification document much like a driver's license. I've witnessed another gentlemen who was denied a passport for child support-related issues. Apparently, the only thing that will get a person from getting a passport is child support and/or tax-related issues.
It used to be that a passport wasn't required for US citizens travelling to the Caribbeans (just a birth certificate and other identifying information), but that is changing with the issuance of passport cards.
You can BEGIN HERE: http://www.travel.state.gov/passport/ppt_card/ppt_card_3926.html
It should be noted that your parole officer or other law enforcement agency may put a restriction on an individual if they are on parole/probation, or for child support, et al.
From http://www.travel.state.gov/passport/ppi/info/info_870.html
A federal or state law enforcement agency may request the denial of a passport on several regulatory grounds under 22 CFR 51.70 and 51.72. The principal law enforcement reasons for passport denial are a federal warrant of arrest, a federal or state criminal court order, a condition of parole or probation forbidding departure from the United States (or the jurisdiction of the court), or a request for extradition. The HHS child support database and the Marshals Service WIN database are checked automatically for entitlement to a passport. Denial or revocation of a passport does not prevent the use of outstanding valid passports.
A felons in general are able to obtain passports but the final authority is the US Government. Even after you obtain a passport or passcard you still have the problem of being allowed to enter other countries. Many will restrict felons from entering (Australia) and some restrict based on the crime(s) involved (Canada, Brittan/UK). Some require that you only stay for a certain length of time and/or register upon arrival.
http://travel.state.gov/travel/cis_pa_tw/cis/cis_1765.html
Keep in mind, we are discussing US passports as I'm sure other countries have their own methods. One thing I do know is that it is difficult for felons to permanently re-locate to another country because nearly all countries require a background check of some sort. The nature of the crime, I'm sure, makes a HUGE difference is who gets accepted or not. I used to know a website that could explain all the relocation requirements for many countries, but I've since lost it. However, you can google up some answers for that subject with something like
(your country) permanent residency.
Again, I'm not an expert. I've done work with prisoners in the past and this is a recurring question.