- The part that I find confusing is: $1-1/Q(n)$, does this mean that we can invert all except a polynomial part which we cannot invert?
This is relaxation from Strong OWF, in which, any polynomial time algorithm will succeed in inverting $f$ on random input with negligible probability.
In weak OWF, any polynomial time algorithm will fail to invert $f$ on random input with non-negligible probability.
- What does this mean in practice?
There is a theorem state as;
Theorem: If there is a weak OWF, then there is a strong OWF. In particular, given a weak OWF $f:\{0,1\}^* \to \{0,1\}^*$, a there is a fixed $m\in \mathbb{N}$ polynomial time input length $n \in \mathbb{N}$ such that $f'$
$$f'(x_1,\ldots,x_m) = (f(x_1),\cdots ,f(x_m))$$
is a strong OWF. In particular this holds for $m=2\,n\,q(n)$.
For a proof see
Therefore, in practice, we know at least one way to construct a strong OWF from weak OWFs.