Let E (for 'election') be the proposition which says de re of each person who will in fact be saved that he or she will be saved. That is, E is the longest conjunction of the form 'John will be saved, and Mary will be saved, and Lois will be saved...' which is true. Let R (for 'reprobation') be the proposition which says de re of each person who will in fact be damned that he or she will be damned.
The doctrine of predestination is the doctrine that God, from eternity, has issued an efficacious decree of election - that is, God, from eternity, effectively chose that E should be true. The doctrine of double predestination states that in addition to the decree of election, God also issued a decree of reprobation - that is, in addition to effectively choosing that E should be true, God effectively chose that R should be true.
Double predestination is much more contentious among Christians than predestination (although predestination is not entirely uncontroversial - for instance, open theists will have to deny it). Many Christians would rather have single predestination, holding that all people are, on their own, bound for hell, and God intervenes to save those he wishes to save, and just leaves the rest alone.
In his Philosophical Theology (1969), James F. Ross proposes the following analysis of omnipotence:
S is omnipotent if and only if for every logically contingent state of affairs, p, whether p or ~p is the case is logically equivalent to the effective choice, by S, that p or that ~p (respectively). (p. 211)
This analysis appears to have the consequence that, if God is omnipotent, then double predestination is true. Both E and R are true contingent propositions, so if God is omnipotent then God effectively chooses that the corresponding states of affairs should be the case.