The definition on Wikipedia of a division algebra $D$ is given as:
Given $a,b \in D$, $b \neq 0$ there exists a unique $c\in D$: $a = bc$ and a unique $d \in D$: $a = db$.
My question(s) are: What exactly does this mean and is it equivalent to this:
For all $a,b \in D$: if $ab = 0$ then either $a=0$ or $b=0$?
Edit Say we assume it's associative.
$\textbf{Edit}^2$: I am so confused! In some places it is defined to be a vector space of a field (with additional properties) and in some places it is defined to be a field in which multiplication does not commute.
No, it is not equivalent. For example, a polynomial álgebra k[X] satisfies your condition but not Wikipedia's.