What actually happened: Bad weather almost forced Ike to postpone D-Day at the last moment, but he decided to let it go ahead on June 6, 1944. Things worked out well, though a sudden, severe channel storm (the worst in a century) wrecked a lot of supply ships, wrecked the Mulberry artificial harbors, made air support impossible, and almost stopped resupply a couple of weeks later (June 19-22).
What might have happened: If the weather had been a little worse on June 6, Eisenhower would have been forced to postpone the landing. That by itself would have caused enormous security problems. Too many people had to know too much in the last day or two before the invasion. Literally hundreds of thousands of people would have to keep their mouths shut and not lose written material they had been given on the invasion. The German spy network in England had been neutralized, but the allies could not be sure there weren't other rings.
There would have been intense pressure to go at the next moment the moon and the tides were right. That would have put the landing around June 18, just in time for the severe storm. That storm, by the way, came up without warning, which is part of the reason it did the damage it did. If the allies had been caught by the storm in the early stages of the landing, with troops trying to establish themselves on the beaches, the troops that made it would have been cut off from resupply and reinforcements for almost four days. Allied air power would have been neutralized. The Germans might well have been able to destroy the invasion on the beaches and destroy airborne forces before they could link up with the main invasion force.
Immediate results: The initial invasion force essentially wiped out--well over 100,000 men killed or captured, including a lot of specialists like the airborne troops. Lots of equipment captured or destroyed, including landing craft, specialized tanks, and other vital cogs in the allied wheel. The artificial harbors that made supply over the beaches at Normandy possible would have been destroyed by the storm on their way over. Bottom line: An allied disaster that would make an immediate second try very unlikely.