What does $\forall$ mean? ProofWiki
What does the upside-down A on this page mean?
$$\forall a,b\in S: a\circ x=b\circ x \implies a=b$$
 A: The symbol $\forall$ means "for all." The symbol $\in$ means "in" or is an element of.
In your context, it means for all $a$ and $b$ in $S$.
A: $\forall$ means "for all", so the statement
$$\forall a,b\in S:a\circ x=b\circ x \Rightarrow a=b$$
means: For all $a$ and $b$ in (elements in) $S$ it is true that if $a\circ x = b\circ x$ then $a =b $.
So saying a bit more: this is just saying that if you have an algebraic structure $(S,\circ)$ like for example a multiplicative group then $x$ is right cancellable if for all $a$ and $b$ if you have that $ax = bx$ then you can automatically conclude that $ a = b$.
Note for example that in the integers minus zero ($\mathbb{Z}\setminus \{0\}$) under multiplication (which actually is not a group) every element satisfies this because you know that if $$a\cdot x = b\cdot x,$$ then the only way that is going to happen is if $a =b$. This is just saying that you can't for example multiply $3$ by anything but $5$ to get $15$.
The reason we can't include $0$ in this example is that for example
$$
4\cdot 0 = 7\cdot 0.
$$
