The equipment of archers and men at arms changed as the war went on, so I will focus on Agincourt. This will be a two part answer, since men at arms/knights and archers had rather different sets of equipment.

Part 1) Men at Arms

To start off with, knights and men at arms and equipped the same way on the battlefield, essentially. A man at arms is a man who can fight as a knight, in full armour. Theoretically they -could- fight mounted but at Agincourt and many other battles they fought on foot. The term 'Men at Arms' includes knights and untitled men at arms, many of whom would still be landowners and many of these would style themselves 'gentlemen.' So the only difference between the equipment of knights and untitled men at arms would be that knights would generally have 'richer' equipment - more decorated armour with finer fittings, better belts and other accouterments. However, wealthy untitled gentlemen also would have 'rich' armour. Tobias Capwell compared full relief funerary effigies with incised slabs and brasses (less expensive funerary monuments) and found that the armour of wealthier men (those who could afford more extensive effigies) showed more stylstic features unique to England and thus was more likely to be made in England. By contrast, many brasses and incised slabs show armour with Italian features, probably because less wealthy men at arms bought cheaper Italian imports rather than custom made English armours.

In 1415 plate armour was undergoing the last phase of its transition from -supplementing- mail armour (as it did in the first half of the 14th century) to being the primary defense (with mail only filling in gaps). Judging by the inventory of the Tower of London armouries, the English Men at Arms were very quick to discard full mail shirts and use only supplemental pieces of mail like skirts, standards (collars), sleeves, voiders and gussets (shoulder/underarm pieces). Older fashioned armours like this one would have a bascinet with a visor (of the snouted type known as 'pig faced' or 'houndskull' to modern collectors). There would be a mail drape, called an aventail hung from the helmet to protect the neck. Newer armours like this one would include some form of plate defense for the throat; there are a variety of styles of this, but most are rigid, made of only a few pieces and restrict the movement of the head more than the old aventail did. These include standing collars of plate, aventails made of 'bands' of plates and even one-piece helmet/plate aventail combinations that project the entire head and neck. The most common as the 15th century progresses is the 'great basinet', which has a separate throat defense riveted in place to the front of the helmet. The visors of more 'up to date' helmets were generally more rounded than the pronounced snouts of older-fashioned helmets.

The shoulders would be protected by simple spaulders. More 'up to date' armours would supplement these with small plates over the armpits (most often round) called besagews. The arms and legs would be almost entirely encased in plate armour, with only the inside of the knees/elbows/armpits exposed - the English in particular favor limb defenses that covered the backs of the legs and insides of the arms. The hands would be protected by gauntlets that had individual plates for each knuckle and a flaired cuff. Modern people call these 'hourglass' gauntlets. Sabatons (foot armour) and greaves both seem to have been worn most of the time, even when fighting on foot.

The breastplate would most likely be a single piece, with a skirt of fauld of plate hoops extending down from the navel level (where the breastplate ended) to the lower hips. In 'older fashioned' armours the backplate might be made of many plates rivetted to a backing, or it might be made in two or three parts fixed together with rivets, hinges or straps. 'Up to date' cuirasses would have a skirt in the back called a 'culet', to match the fauld in front. More 'old fashioned' armours would have their cuirass covered with fabric or leather, but more 'up to date' armours would have the metal exposed. Around this time English men at arms mostly stopped being depicted wearing surcoats/jupons over their armour.

Armour was decorated with brass borders riveted onto the arms and legs of the armour, and sometimes with a kind of circlet-like piece of headwear called an orle, which seems to have sometimes been gilded.

For weapons, English men at arms fighting on foot would be armed primarily with polearms or 'short lances' - spears, basically. The most popular of these was probably the pollaxe, which was a 4'-6' with a spike at the top and some combination of axe blade, hammer and spike on the head. The Tower Armouries from the early 15th century record thousands of 'short' lances, which seem to have been spears used in two hands by dismounted men at arms - they were used extensively in the 14th century and if the Tower Armoury records are any indication, were still present in the early 15th. However, their persistence, unused, in the Tower inventory may indicate that by the early 15th century they were no longer being actively issued, and instead sat in storage.

Swords were carried and used - they were less useful against armoured men, but were very versatile and extremely useful against unarmoured opponents. Most early 15th century swords have a very 'pointy' shape - Eward Oakshotte classified these as sword types XV through XVIII. Older authors sometimes say that these swords were designed to pierce plate armour with thrusts, but this is misleading - swords cannot, in general, penetrate plate armour. However the 'pointier' shape would help when sliding swords into the gaps in armour forcing them through. Swords could be designed exclusively for one-handed use, or for use in either one or two hands ('longswords'). Here is a sword of Henry V. The English men at arms do seem to have used two-handed swords, though the only surviving example is from the middle of the 15th century. Interestingly, swords were occasionally used couched during mounted charges, like a lance - which is another demonstration of how flexible they were as weapons. On occasions when English Men at Arms fought on horseback in the early 15th century, they would be armed with a lance that was essentially a 11' long spear - it did not yet have the swelling around the hand grip that later 15th century lances have.

Sources:

Tobias Capwell - Armour of the English Knight 1400-1450 - details on armour. This book rewrote scholarship on 15th century English armour, and is essential reading on the subject.

Edge and Paddock, Arms and Armour of the MEdieval Knight - General background on pollaxes

Tobais Capwell - Masterpieces of European Arms and Armour at the Wallace Collection - notes on two-handed swords and pollaxes

Ewart Oakshott - Records of the Medieval Swords - notes on swords

Alan Williams - The Knight and the Blast Furnace - notes on weapon's effectiveness against armour

Thom Richardson - The medieval inventories of the Tower Armouries 1320-1410 - notes on mail shirts versus supplemental pieces of mail armour