Here is an example of per impossibile reasoning from Thomas Aquinas, De Veritate, q. 1, art. 2:
Even if there were no human intellects, things could be said to be true because of their relation to the divine intellect. But if, by an impossible supposition [per impossibile], intellect did not exist and things did continue to exist, then the essentials of truth would in no way remain. (tr. Mulligan)
The gist of the passage is this: it is impossible that God not exist; but if he did not exist, then truth would not exist. Truth depends for its existence on a being whose nonexistence is impossible. To highlight this dependency relation, Aquinas bids us consider what would happen if, per impossibile, God did not exist: in that impossible counterfactual situation, truth would not exist either.
Doesn't the intelligibility of the above passage and thus of per impossibile reasoning show that there is a distinction between conceivability (thinkability without apparent logical contradiction) and possibility and that conceivability does not entail possibility? The nonexistence of God is clearly conceivable — if it were not, the passage above would be unintelligible. But the existence of God is also clearly conceivable. So if conceivability entails possibility then the upshot is that the existence of God and the nonexistence of God are both possible. From this it follows that God is a contingent being. But then the view that conceivability entails possiblity forecloses on the possibility that Aquinas is right and that God is a necessary being, one whose nonexistence is impossible. Whether or not Aquinas is right, it is as least epistemically possible that he is. Therefore, conceivability does not entail possibility: the conceivability of X is consistent both with X's impossibility and X's necessity.