RealityChuck has the process described quite well.
Don’t mistake institutional accreditation (which is also called “regional” accreditation, because it’s handled by regional organizations) for program accreditation. In addition to institutional accreditation (a MUST for any school who wants their students to be eligible for federal financial aid dollars), individual programs (particularly professional ones) may seek specialized, field-specific accreditation. These accreditations (or lack of them) play no role in an institution’s regional accreditation. Perhaps this is where Daylon’s conception comes from.
Yes, some institutions don’t seek institutional accreditation. Maybe they think they won’t ever qualify (too few resources, for example), or maybe they want to do things their own way and are happy to declare “independence” from oversight and from federal $$ (Hillsdale College comes to mind.) Students from these schools may have problems transferring credits or entering graduate school.
It’s much less unusual for specific programs to eschew field-specific accreditation. Sometimes the schools simply don’t agree with the the values of the accrediting association. For example, U-M’s Ed School doesn’t agree with NCATE standards, so they don’t bother. Our teachers still pass their exams and still get jobs, so it’s no biggie.
Now, as for why students go to these places with no real accreditation… some students are surely unaware of the significance of accreditation. Schools can also announce they are accredited by their own sort of specialized body (like, say, the “National Board for Internet Degrees”) and students don’t know the difference. And some students want a certain something so badly (say, for example, a strong denominational affiliation) that accreditation is relatively unimportant to them.