What does it mean to say "a divides b"

Given two integers $a$ and $b$, we say $a$ divides $b$ if there is an integer $c$ such that $b=ac$.
Source.

This is what $a$ divides $b$ means. The shorthand notation is $$a|b$$.

In your example, $$a|a^2\iff a\leq a^2$$ since by definition there exists $c$ such that $a^2 = ac$, namely $a = c$.

We say $a$ divides $b$, denoted by $a | b$, if $b$ is a multiple of $a$ (ie, $b$ is an integer multiple of $a$). Equivalently, $a |b$ iff $b=ka$ for some integer $k$.

To remember what "$2$ divides $6$" means, perhaps you can remember the phrase "$2$ divides $6$ into $3$ parts". Hence, $2 | 6$.

Note that $2 | 0$ because $0$ is an integer multiple of $2$: $0 = k2$ for some integer $k$. Just take $k=0$.