If you choose the second mode, no new blue or black spell may be cast by an opponent that turn targeting a permanent you control after Display of Dominance resolves.

Permanents you control may be the targets of abilities from blue or black sources controlled by your opponents.

Keep in mind that an Aura spell targets the permanent it will enchant (but an Aura on the battlefield doesn’t target the permanent it’s attached to).

If you choose the second mode, and if a permanent you control is being targeted by a spell when Display of Dominance resolves, nothing happens right away. When that spell would resolve, its color is checked. If it’s blue or black, that permanent will be an illegal target for that spell and won’t be affected by it. If all that spell’s targets have become illegal by the time it would resolve, it’s countered.

If you choose the second mode, Display of Dominance will affect any permanent you happen to control at any point during the rest of the turn, not just permanents you control as it resolves. That’s because it doesn’t grant an ability to those permanents; rather, it affects the game rules and states something that’s now true about those permanents.