Non-airborne "leg" infantrymen, artillerymen, combat engineers, and most other American soldiers wore the M1939 service shoe during WWII. This was an ankle-length leather boot. The shorter boot was easier to march and move in. However, the low-topped boots could easily get flooded with mud, sand, etc.
During WWI and the early 1900s, the typical solution to this problem was to wrap strips of frabic called puttees around the lower calf and ankle. American Doughboys and British Tommies both fought WWI wearing puttees. However, puttees weren't an ideal solution. They were finicky to put on, they easily unraveled, and they wore out pretty quickly.
So, American soldiers were issued with M1938 gaiters, a canvas sleeve was laced around the ankle and buttoned under the boot. This improved ankle support and kept debris and muck from getting in the top of the boot.
Parachute infantrymen wore distinctive, high-topped jump boots. At the time of the D-Day landings, the all-leather Corcoran jump boot was by far the most common. These boots went above the paratrooper's ankle, giving him better support when he landed with 100+ pounds of equipment. Since the higher-cut boot also prevented the ingress of mud and the like, paratroopers didn't need gaiters. As a paratrooper-only uniform item, jump boots quickly became a mark of pride for the paratroopers.
During the Normandy invasion, the glider infantrymen who served in airborne divisions were not issued with jump boots (nor were they given the same danger pay as paratroopers).