At dawn on 23rd October 1415, both King Henry V, the English king, and the Duc d'Albret, the French Commander laid out their forces near their respective camps. To start, the lines were a little over a mile apart, a little to the south of the French village of Azincourt (which is in the present-day departement of Pas de Calais). We, of course, had to bastardise the name, because that's what we English do best. The plain between the armies was a gently rolling field, freshly plowed and planted, about 900 yards wide. It had been raining continuously for two weeks, and the field was a sea of mud.
The French had two very dense lines of armored foot soldiers with genoese crossbowmen and bombards between. Mounted knights guarded the flanks and formed a reserve in the rear. d'Albret's plan was to use the bombards to cut the English lines into smaller sections that could be handled individually. Unfortunately, everybody (including d'Albret!) wanted to be in the front line. It got so dense that the bombards couldn't be fired, as they would hit more French than English. They actually were fired once, to no effect.
The crossbowmen were equally ineffective. Being mercenaries, their desire to fight was not huge, and was helped only by a small pot of gold. Due to either their work-shy nature or plain stupidity, the crossbowmen of the French force allowed their bowstrings to get wet overnight before, and thus were almost useless.
Henry laid out his forces in the traditional English fashion, with men-at-arms flanked by wedges of archers in the centre, protected by large pointed stakes. (Horses won't charge at big pointy things.) The archers at the ends of the lines were positioned forward from the rest of the troops to give covering fire along the main front. This is an excellent defensive position, but it gives very little scope for attack. In the woods on the flanks were the cavalry, under the Dukes of York and Clarence.
After the forces were arranged, they sat and stared at each other for four hours. The English had no desire to attack, and the French were presumably not pleased at the idea of wading through a mile of mud.
About 11 AM, as some of the French were sending their servants back to camp to bring lunch, Henry decided to force the issue. He ordered his troops to move the line forward, and to reset the positions within extreme longbow range from the French lines. He didn't have enough men-at-arms to form a reserve or to guard the camp. This was to have dramatic consequences later on.
As Henry had planned, the first volley of arrows goaded the French into attacking. The first attack was from the mounted knights on the flanks of the French position, intending to overrun the longbowmen protecting the English flanks. It was a disaster. While an English arrow would not normally penetrate a knight's plate armor, a horse cannot carry enough armor to be effective. Wounded horses threw their riders into the mud and trampled through the close-packed ranks of French foot soldiers. They also churned up the mud in front of the English positions, making things more difficult for future French attacks.
The main French attack was from the first line of men-at-arms. Unfortunately, everybody tried to push their way into the first line, including Constable d'Albret. As they marched toward the English, their line was squeezed together by the narrowing field, until they were so close together that they couldn't lift their arms to use their weapons. However, even with the mud and the crowding, the shock of the French men-at-arms hitting the English line was terrific, throwing the lines back for several yards.
It was, however, ineffectual. Despite some terrific fighting, the English line was never in any serious danger. While men-at-arms in plate armor are normally quite mobile, the combination of the mud and the crowding made them almost helpless. The English simply knocked them down, to drown or suffocate under fallen bodies.
The second line of men-at-arms followed the first. Now, however, there was the added complication that the English positions were blocked by a wall of bodies. The second line had no better luck against the arrows, mud, and English men-at-arms than the first.
After the collapse of the second line, the English common soldiers started in on the traditional battlefield activity of taking prisoners for ransom and stripping the armor and jewelry from the dead. However, the remaining French forces, both the survivors of the first two lines and the entire third line, plus the crossbowmen, easily outnumbered the English. As the counts of Marle and Fauquembergues tried to rally the French for a third attack, Henry gave the order to kill the prisoners. This removed the risk of the prisoners turning on their captors and freed their guards for duty elsewhere.
At roughly the same time, a group of French knights cut through the woods and attacked the English camp. In Shakespeare, the raid on the camp was Henry's reason for ordering the prisoners killed; I suspect that it was a later justification. Remember, the murdered prisoners represented a very large amount of ransom money, which Henry needed very badly.
The attack of Marle and Fauquembergues was defeated with no particular effort. Their charge (in which both of them were killed) was the last offensive action that the French mounted.