The first amendment is the freedom of speech / religion and the ability to express your thoughts in public without getting in trouble for doing it
The case was about a teenager posting a blog entrance and was punished for it, she wrote how the school officials got mad about it and forbidden her to run for class secretary again.
So Avery and a bunch of others came to School with a t-shirt on that said team Avery on them and they were prohibited to wear the again
“Avery does not have a First Amendment right to run for a voluntary extracurricular position as a student leader while engaging in uncivil and offensive communications regarding school administrators.”
In the end she lost the case but won the ability to wear the t-shirts because the court said it was wrong of the School to say they were election material and said the students should be able to be enfolded in a non-disruptive and non-violence political speech on school grounds.