Let $G \in \mathbf{Grp}$, $H \leq G$, $G/H := \lbrace gH: g \in G \rbrace$. We can then introduce group operation on $G/H$ as $(xH)*(yH) := (x*y)H$, so that $G/H$ becomes a quotient group when $H$ is a normal subgroup.
But why do we only work with quotient groups by normal subgroups? If we introduce the notion of left quotient group in the above manner, how much good properties of a quotient group do we lose?

I cannot comment yet, but I thought I would elaborate on what is meant by being well-defined.

In your definition of the multiplication in $G/H$, you take cosets $xH$ and $yH$ and multiply by setting their product equal to $(xy)H$. But this means that you have chosen $x$ and $y$, and there will in general be other choices of $x$ and $y$ that give the same cosets. So if you have $xH$ = $x'H$ and $yH$ = $y'H$, you want to be sure that you also have $(xy)H = (x'y')H$, and it can be shown that this holds for all elements of $G$ iff $H$ is normal in $G$, so this is the only case where this multiplication is well-defined.
