This decision reversed a previous ruling from 2000, involving New York University.
The decision does not affect all public universities in the USA, as some State governments have laws which expressly permit graduate students to form unions for the purpose of collective bargaining. According to the AFT link below, at least 13 states have recognised the right of graduate students to form unions.
The issue of graduate student unionization has been hard fought for at least the last 25 years and seems certain to continue to be contentious for the forseeable future.
Update [2004-8-3 6:19:29 by overconvergent]:
The American Federation of Teachers called the decision "outrageous".
According to the Chronicle, there are other, similar cases at private universities which will come before the NLRB soon.