American Bar Association

American Psychological Association a major professional organization in Psychology that accredits graduate school programs.

Distance Education And Training Council is recognized by the US Department of Education

National Council for Accreditation of Teacher Education a Specialized organization that accredits undergrad and graduate programs in education.

National Engineers Examining Board

This is my opinion:
Going outside the protective circle of regional accreditation is usually not a good idea and presents a complicated picture. There is where things get complicated! List of State Agencies that approve or register higher education institutions.
       (I put this here as general information, as this has been an issue for some students who have emailed me.) It is far better to go to a regionally accredited school (AND as I say in the introduction), there are professional organizations that accredit graduate programs (such as the ABA or APA) that are even better than regional accreditation. But, normally schools that have such advanced accreditation will also have regional accreditation.
      Now, while I'm at this let me mention one curve ball: church degrees. If you want to be a minister in a particular religion you should go with a school that is accredited by that religion, or that is accredited by an organization supported by that religion. Now, religious accreditation organizations that are widely recognized will usually appear on the Department of Education list. However, it is a fact that if you want to be a minister in a particular faith that requires a divinity degree, you will have to go to a school accepted by that faith. It is also a fact that some employers, such as those that hire hospital chaplains, may not recognize a degree from a school that is accredited only by an obscure or not-widely known church (although such religions, such schools, and such churches may have great merit). If you want that kind of "chaplain" or "pastoral counselor" job outside of the church itself, it is a better bet to go to a religious school accredited by an organization on the Department of Education list. Check with a prospective employer, such as a hospital that employs chaplains, to see what type accreditation they accept.
      Apart from all that, if you even consider a college does NOT have regional (or valid professional) accreditation it should, have accreditation from one of the sources recognized by the Department of Education(Valid sources of "professional" accreditation, in case there is doubt, should also appear on the Department of Education list).
      On a lower rung, but still generally on a rung, are schools that are state approved/registered institutions. Here is my point: If an institution is as a minimum not state approved/registered it is possible that it doesn't really exist or is not legal. Again, it is best to go with a regionally accredited school.
      Now, let me add this also, there has been criticism of regional accreditation bodies which says the they are conservative and therefore do not often facilitate innovation, they are also said to be bureaucratic and slow. It is probably true that there are non-regionally accredited schools that can give you a great education. The problem for me is that you may end up well educated, but your degree may not be recognized. And therefore you may not benefit the way you would like. Never-the-less, I believe that valid institutions exist that do not have regional accreditation. Therefore, I cannot say NEVER consider such an institution. However if you value recognition as well as education ... if you value possibly being able to transferee your credits to another good institution if you have a mind to ... if you are going to go to the trouble and expense of getting a valid education anyway ... if you don't have the time or resources to search for the "hidden jewel in the rough" ... my opinion is go with the sure fire real accreditation, accept no substitute.

Western Interstate Commission on Higher Education an organization involved in the debate regarding distance education.

Council for Higher Education Accreditation An authoritative non-government source of information on college and university accreditation.