The effect of this ruling hinges on what a judge would define as "legitimate pedagogical reason". I was in high school and writing for the newspaper when this ruling came out, and we had an issue at our school where the Principal wanted to censor out stories that dealt with topics they felt were too controversial, such as abortion or teen drinking and drug use.
My journalism adviser threatened to resign over it, as he felt it was a first amendment right, and that this was not a pedagogical issue, but a personal one for the Principal. The Principal argued that since the newspaper was funded with public dollars, the same rules as a private newspaper did not apply, and the concerns of parents who did not want to fund stories of that nature were valid. In the end, the School Board used Hazelwood as a reason to back the principal and my adviser resigned.
I think the practical result of the ruling was that more and more schools simply do not have school newspapers or journalism classes.
This was a Supreme Court decision from back in 1988 that helped to clarify what First Amendment rights high school newspapers have.
In the case, the Supreme Court said that school newspapers are not as highly protected from censorship as regular newspapers are. They said that school papers can be censored if authorities have a legitimate pedagogical reason for doing so.
So, a school does not have to allow speech that goes against its "basic educational mission."
It's also somewhat important to note that the Court ruled that the school paper in question had not been used as a "public forum" in the past and so the school retained the right to censor it. If it had been a public forum, the school would have been less able to censor.