Do American students enroll in unaccredited universities? In countries with central governmental control over higher educations, universities normally should be accredited by a governmental body to establish a university and enroll students.

However, in the United States (name if other countries have similar system), establishment of a university is different from accreditation. Therefore, a university officially starts to enroll students (correct me if I am wrong), then apply for accreditation by local accrediting agency.

During this period (establishment and accreditation), will students enroll in a university which has not been accredited yet?
There are two very different cases that should be distinguished.

You might have a brand-new university that is seeking to become accredited, but needs to be operating (with students) for a few years before accreditation is awarded.  An example is Olin College which enrolled its first class in 2002, and received accreditation in 2006-2007.  The first few classes of students do have to consider the risk that the institution might never receive accreditation, or might not establish a good reputation, or might shut down completely before they graduate.  So the school has to do a good job of building a reputation fast.

Prospective employers of the first students could conceivably be bothered by the fact that the institution is not accredited; however, reputation typically counts for much more than accreditation per se.  Also, in the case of Olin, note that accreditation came through just shortly after the first class graduated, and after that, nobody is really going to check whether they got their degree before or after accreditation.

The other case is universities that are not accredited and do not want to be.  There is nothing illegal about this; in the US, accreditation is a purely voluntary procedure.  However, most such institutions have a poor reputation or none at all.  A few have genuine educational programs but prefer to remain independent of the accrediting process for philosophical or religious reasons.  Others are mere "diploma mills" with no real educational product whatsoever.  Students should be very cautious about choosing such non-accredited institutions, but they are free to do so if they want.  Employers are likely to view such degrees with skepticism; however, this would be based more on the institution's lack of reputation than its lack of accreditation.