A lot can be told on all the aircraft here. Including on the gliders. As a fact, books have been written about all of them. As this is not a book, it will be less detailed. But a nice start to learn a bit more.
The Germans were the first to use gliders in a combat operation. That was on the famous attack at Fort Eben Emeal in Belgium. The attack was a success and the Germans kept their glider program secret.
Nevetheless, the allied started their program out ot nothing, but came out with something good.
Waco CG-4A

Dispite what people might think, this aircraft has nothing to do with Waco, Texas, the town. Waco was an aircraft company in Troy, Ohio (Weaver Aircraft Company). The aircraft was designed by Francis A. Arcier. The CG-4A was constructed of fabric-covered wood and metal and was crewed by a pilot and copilot. It had two fixed mainwheels and a tailwheel.
Sixteen companies were involved to built a total of 12.916 gliders during the war. Subcontracters were as many as that, including companies who actually built pianos.
The CG-4A could carry 13 troops and their equipment. Cargo loads could be a 14ton truck (i.e. a Jeep), a 75 mm howitzer, or a 14ton trailer, loaded through the upward-hinged nose section. C-47s were usually used as tow aircraft.
The USAAF CG-4A tow line was 1116-inch-diameter (17 mm) nylon, 350 feet (107 m) long.
The CG-4A pickup line was 1516inch- (24 mm)-diameter nylon, but only 225 ft (69 m) long including the doubled loop.

The first use of the CG-4A was in the Sicily invasion. The glider operation could not be considered a success as most gliders failed to reach the Landing Zone (LZ) due to being cut off too far from the coast lines. This failure did put presure on the glider program and an accident in St. Louis, where civilians on a glider ride were killed when one of the wings came off, did not help either.
It was an excersise with arranged by Mike Murphy that saved the glider program, as gliders landed without a sound, and in the dark, at the exact spot where they were suposed to land.
A Waco CG-4A glider and a C-47 at South Plains Army Airfield (SPAAF) at Lubbock, September 1942.
After the Sicily  invasion, the Waco CG-4A gliders were used in large numbers in Normandy, Southern France, Holland and Germany. Beside that, the Bastogne resupply missions included gliders as well.
For Normandy, the main problems were the small fields and the high hedgerows and trees seurrounding those fields. In Southern France the grape fields caused some problems.
Holland had some soft ground issues, especially in the Groesbeek area. But overall, the use of the Waco CG-4A glider was succesfull.
Left: C-47 towplanes of the 88th Troop Carrier Squadron / 438th Troop Carrier Group are lined up to tow the gliders to Normandy. This is part of the Elmira mission, June 6 1944.

Left: One of the gliders, of an unonown unit., cracked up on landing.  
Left: Glider being loaded with a jeep. This is at Aldermaston, home of the 434th Troop Carrier Group.
Above. A glider on the landing zone in Holland is greeted by Dutch civilains.
Left: WacoCG-4A gliders in a field in southern France. These are most probably 436th Troop Carrier Group. (Z. Graves collection)

Below left: A Glider of the 27 December flight has landed on the LZ near the positions of the artillery units. It is believed that this glider was piloted by F/O Claude Berry, 91st Troop Carrier Squadron, 439th Troop Carrier Group.

Below: A glider at LZ-N at Wesel. German small arms fire and anti-aircraft fire was very heavy during this operation. More Troop Carrier men died on this day then any other during the war.
Airborne Troop Carrier - Articles
Interior of a glider, probably taken during a test flight. Note the guy relaxed reading LIFE magazine. (H. Thapp collection)
Waco CG-13A

As follow up glider for the CG-4A, the CG-13A was developed. This glider was shipped to Europe and, like with the C-46, the 313th TCG was the first to become equipend with this type of glider. These gliders were not used in any combat operation in europe.
The glider could carry  30 troopers, and, like the British Horsa glider, a jeep with trailer/gun.
Left: side view of CG-13A. Right: Inside view of CG-13A. (C. Gwartney collection)
Airspeed Horsa

This was a British built glider. As with the americans, the British had to develop their glider program ou of scratch. The glider could carry 30 troops. Or a jeep with trailer/gun. This was a big difference to the american Waco CG-4A. The CG-4A had to land in pairs to keep jeep and trailer/gun together. The British Horsa could land them at once, which was an advantage in assembling.
The Horsa glider was used by the US Troop Carrier units as well. Upon arrival in Britian, US Glider Pilots started to fly with this bigger planes then their familiar CG-4A.
The Americans used the Horsa glider only once in combat. That was on the invasion of Normandy. The area of landings was not well suited for gliders, and the bigger Horsa gliders were in disadvantage to the CG-4A. With crashes, casualties from the Horsa gliders were considered higher. The logic is that more men were in these gliders. However, it ended the use of this glider by the Americans in combat operations. For training, the gliders were still used, as is evident by a 438th TCG accident with such a glider in December 1944, killing pilots and 17th Airborne Division occupants.
The British used this type of glider in all their Airborne operations. During Market Garden, the results of these gliders were perfect.
Normandy related photos showing the Horsa glider with the IX Troop Carrier Command. Left top: F/O Ed Short of the 435th Troop Carrier Group stands beside the Horsa glider he is going to fly into Normandy.
Top: A 90th  Troop Carrier Squadron towplane (one of the four squadrons of 438th  Troop Carrier Group) tows a Horsa glider from Greenham Common. This is the Elmira mission.
Left: One of the badly mangeld Horsa gliders in Normandy. Small fields and enemy fire resulted in bad crash landings.
Below: Three Horsa gliders that made a good landing. Two have their tail removed. This was the way to unload vehicles. Note the British roundel visible on the wing! Three parachutes are visble in the same field.  
Airborne Troop Carrier machines - C-109
Airborne Troop Carrier machines - C-46
Airborne Troop Carrier machines