What does the | symbol mean in $3 | n^3 + 5n + 6$?

Usually $n\mid m$ means that $n$ divides $m$ (or $m$ is a multiple of $n$). For example $3\mid 6$ but $3\not\mid 7$.

Writing $a | b$ means "$a$ divides $b$" or "$b$ is a multiple of $a$" or "$a$ is a divisor of $b$", which are all different ways of saying the same thing.