Let's suppose that we have two normal distributions, and a sample is taken from each (of sizes $n_X$ and $n_Y$ respectively).
$X \rightarrow N(\mu_X, \sigma_X ^2), Y \rightarrow N(\mu_Y, \sigma_Y ^2)$.
We want to test: $H_0$: "$\sigma_X ^2= \sigma_Y ^2$" against $H_1$: "$\sigma_X ^2 \neq \sigma_Y ^2$".
Depending on whether $S_X ^2 > S_Y ^2$ or otherwise, we take the decision variable as (supposing the former case):
$$F = \frac{S_X ^2}{S_Y ^2} \rightarrow \operatorname{Fischer}(\nu_1 = n_X -1, \nu_2 = n_Y - 1)$$
Everything is fine until here (though I'm not quite sure if this is the correct formula). Anyhow, how would I find the critical region? Supposing that the risk is $\alpha$.
By "risk", I take it you mean what I would refer to as "significance level".
The right-half of the critical region is the values greater than $F_{R}=F_{n_{X}-1,n_{Y}-1,\alpha/2}$, where $F_{R}$ is defined to be the value such that $P(X>F_{R})=\alpha/2$ if $X \sim \text{Fischer}(n_{X}-1,n_{Y}-1)$. This can usually be looked up in tables.
What usually cannot be looked up in tables is the corresponding value $F_{L}$ such that $P(X<F_{L})=\alpha/2$. Fortunately, this can be found by the formula $$F_{L} = \frac{1}{F_{n_{Y}-1,n_{X}-1,\alpha/2}},$$ taking care to note that we have swapped the degrees of freedom in the denominator.