Rather than acting now, you delay until a later time. The rest of your turn doesn't happen yet. Instead, you're removed from the initiative order. You can return to the initiative order at the end of any other creature's turn. This permanently changes your initiative to the new position. You can't use any reaction abilities until you return to the initiative order. If you Delay for an entire round without returning to the initiative order, the actions from the delayed turn are lost, your initiative order doesn't change, and your next turn occurs at your original position in the initiative order.
When you use Delay, any negative effects that normally occur at the start or end of your turn occur immediately when you use the delayed action. Any benificial effects that would end at any point during your turn also end. The GM might determine that other effects end when you use Delay as well. Essentially, you can't Delay to avoid negative consequences that would happen on your turn or to extend beneficial effects that would end on your turn.