One of the first things to keep in mind about the soldiers serving at Agincourt is that they were not inherently "infantry." By 1415, most archers hired for military campaigns were supposed to be mounted, as were the men-at-arms. They would typically dismount to fight in a proper battle, but were mounted for scenarios such as raiding or reconnaissance. They might also remount in the middle of a fight to perform a flanking maneuver at a critical moment, as a mixed group of men-at-arms and archers did at Poitiers. Archers would essentially be serving multiple roles: as missile troops, as light cavalry, and light infantry. Terms like "infantry" and "cavalry" are often misapplied. While in later eras, these were stricter descriptions of what a unit did, in the medieval period, troops were much more adaptable. A high proportion (or even majority) of horsemen were critical to the mobility and utility of medieval armies, which were typically quite small.

I emphasize the fact that archers rode horses to battle (if not necessarily in battle) because that indicates another important feature of the average soldier at Agincourt. Archers of the later 14th and 15th centuries were often described with the term valettus, which was used as the equivalent of the term "yeoman." Valettus is of course similar to our modern valet and has the same connotation of service to an individual. Archers were quite often men employed in the household. The kings of England had a personal guard of 24 mounted archers, some of whom were also personal attendants. Some archers are listed on muster rolls with surnames like Baker, Carpenter, Armourer, and Bowyer, which presumably reflect their professions (others, like Millerson, reflect their origins). There was quite a variance in terms of personal wealth, even within small units. Richard II's bodyguard contained some men who were well-off landowners and some whose personal wealth amounted to a pair of cows. For men in service to a particular household, daily life consisted of their regular jobs (which, as we've seen, might be a variety of trades). The Yeoman of Chaucer's Canterbury Tales, for example, is employed as a forester for the Knight's lands, but is accompanying the Knight and the Squire as their bodyguard while on pilgrimage. The Yeoman would have also accompanied his masters to war, presumably along with other men of the retinue.

Archers not attached to a household might find their way to military service by contracting with an individual man-at-arms. English armies of the Hundred Years War were raised by contracts of indenture. At its most simple, a soldier would sign a contract with a retinue to serve for the duration of the campaign. The system of contracts could get quite labyrinthine. Free companies might hire themselves out as subcontractors to a knight looking to acquire a few more archers for his retinue. Individual men-at-arms often contracted with a single archer. The man-at-arms John Hampton and his comrade, known to history only as "Hampton's archer," was recruited for a naval campaign in 1374. Other men-at-arms in that unit hired their relatives to accompany them as archers. These relatives might have been poorer cousins, nephews, or possibly even younger brothers. Poorer gentry families might have their eldest son equipped as a man-at-arms, with any additional sons being outfitted as archers. A Thomas Mascy who was an archer employed by Richard II in 1398 is presumed to be the same Thomas Mascy listed as a man-at-arms two years later. Archers were paid relatively well for men of their rank and station, with opportunities for acquiring loot while on campaign. As the research of Gary Baker shows, we cannot presume that all archers were well-off yeomen, but yeomen were certainly well-represented in the ranks of the archers.

Just like the men-at-arms, the archers were capable and experienced troops. As Anne Curry puts it,

Do these long-serving archers have a claim to military professionalism? For the late fourteenth century this is a difficult question to answer because warfare in this period was generally recurrent rather than continuous, and the muster evidence has large gaps...As we move into the fifteenth century, however, we are on firmer ground in claiming that many archers were serving as professional soldiers, in the sense that soldiering constituted a full-time, year-round form of occupation for a large number of those serving in Normandy and the pays de conquête. (From The Soldier in Late Medieval England, p. 171)

Certainly there is enough evidence to say that a large percentage of them were professional soldiers. Even when not fighting directly for the English crown or stationed in garrisons, many would go abroad to fight as mercenaries. There were English archers fighting in Portugal, in Italy, in Burgundy, and even as far as Transylvania. They were generally well-equipped. Besides their bows, they carried swords, daggers, and bucklers, but also might use poleaxes or some other kind of shorter staff weapon. At Agincourt, some archers famously fought with mauls, but it is unclear whether this refers to mallets they used to pound in defensive stakes or actual warhammers designed to fight armored men. Possibly the answer is both. Their armor might consist of a quilted cotton or leather jack at minimum or at best a mail coat or brigandine. If they had helmets, they were open faced (an obvious necessity for missile troops).

Their time spent on a full-scale battlefield would have been rather minimal. Raids, sieges, and smaller actions were the main form of combat in the middle ages. Pitched battle was risky and therefore rare, although some commanders did attempt to force decisive engagements when they felt they had the advantage. On the campaign trail, mounted archers would do the same tasks as men-at-arms (riding around, looting supplies, figuring out where the enemy was and what he was doing). Although a siege might provide opportunities for fighting during an assault or if the enemy sallied forth, sieges were filled with manual labor and tedious waiting. Pitched battles like Agincourt would be uncommon and all the more brutal and horrifying for their rarity. On the battlefield, archers would be positioned on the flanks and (ideally) protected by field entrenchments. Natural terrain like hills and hedges could be supplemented with defensive trenches and wooden stakes. Behind these defenses, archers could safely harass the enemy. The purpose of the archers was not to slaughter the enemy outright with their missiles, but to disrupt enemy formations, hamper their movements, and provoke them into making rash decisions. The men-at-arms would take the brunt of the disrupted enemy assault while the archers dropped their bows to fight hand-to-hand. When commanded well, this could result in spectacular victories, but English archers were no less vulnerable to bad tactics as anyone else.

Sources:

The Soldier in Late Medieval England, by Bell, Curry, King, and Simpkin.

Arms, Armies, and Fortifications in the Hundred Years War, by Curry and Hughes.

Agincourt, a New History by Curry

It would help to narrow down your particular field of interest. Did you have a specific period of years, a nation / culture, and social class in mind?