Since you specify “small village” in your question, I’m going to confine my answer to that because it would vary a bit if you lived in a large village, or, especially, in a town, or were a tenant working a monastery’s land.

Let’s start with the ecclesiastical division in which your village is contained: the parish (and also realize that a rural parish could hold more than one village). The parish was a demarcated property, the spiritual and legal entity over which your priest presided. There were around 9,000 of these in England in the 14th century. It was the locale where a priest ministered to your spiritual needs, but it was also a social and, especially, an economic entity. The “village priest” was entangled in all of these. Now, about that “village priest.” Until the mid-century Black Death decimated clerical numbers, all but the poorest parishes would have two or more priests serving them. But these priests came with several titles and responsibilities. First and most importantly, there was the rector of the parish, the man who was responsible for the “care of souls” (cura animarum) in your parish. This means he administered five of the seven sacraments to you: baptizing you, hearing your (usually) annual confession, giving you your (usually) annual communion, blessing your marriage, administering the last rites to you, and burying you. (You were supposed to receive the sacrament of confirmation at a young age, but only bishops could administer this and it’s not clear how regularly rural parishioners received this. Likewise, only a bishop could ordain someone to the priesthood.) The rector was also obliged to preach to you several times a year. In return, he was entitled to your tithe, the tenth part of everything you earned or produced (i.e., wages, but more usually the crops you grew or animals you owned and bred). One Glastonbury rector tried, in vain, to claim the tenth child of one of his parishioners! He also received customary payments—oblations or altargage—for services such as baptizing you, or giving you blessed bread (not the consecrated Host) at the end of Mass, or “churching” you after you gave birth. (Technically you weren’t buying these services, that would be the crime simony, but making them as a voluntary offering.) In blunt terms, the average parish could be a money-maker for its rector though many parishes eked by, like the fictional one administered by its rector, Chaucer’s “poure parsoun.”

The hitch here is that by the 14th century the rector of your parish may have been non-resident in your parish, may not, in fact, even have been a single individual but instead a legal corporation like a monastery, convent, or college. In that case, the absentee rector hired a vicar to reside in the parish and perform these duties for him (or, if a corporation, them). Rectors paid vicars a (hopefully) decent annual wage while the bulk of the parish’s income went to the absentee rector. If the rector didn’t pay him well enough, church authorities might create a “perpetual vicarage,” which gave the vicar the same rights and income as its vicar. By the thirteenth century, probably half the parishes in England were vicarages or one type or the other.

The rector or vicar had assistant priests (chaplains) to aid him administering the sacraments, saying mass, and looking after his parishioners. They too were salaried, and often not that well. As well as his other duties, an assistant might offer some rudimentary education to the boys of the parish, especially those who showed some aptitude and could be groomed to become priests. By the end of this period there were also private priests, called “chantry priests, hired by wealthy families or local guilds, to say private masses for the souls of departed members. By the early 15th century, these men were even more likely to spend their free time teaching school. In short, there were several kinds of priests who usually lived in your parish, not to mention men and boys in minor orders, not ordained full-fledged priests, but able to perform certain clerical duties.

And, yes, you saw your priest outside of church. He lived in the rectory or vicarage, usually made of the same sort of materials as your own house. If he was assistant priest and the rectory wasn’t big enough, he may well have lived with you in a spare loft. If there were enough assistant clergy, they may have lived together in the same house. He usually worked side by side with you doing ordinary farm work because he owned—more like rented—land just as you did. (This was a source of friction in that he worked the same land as you did but also got ten percent of what you produced through tithing, or at least the rector did, and he didn’t lose a tenth of it to the tithe.) But it’s well worth remembering that the clerical life was the one profession that was truly open to talent in the Middle Ages regardless of social class. Your parish priest might have come from some village down the road or even be a boy from your own village you played with as a child, or he might be the son of a nobleman.

While you rubbed shoulders daily with the priests in your parish, they were a class set apart. Increasingly “professionalized” from the 13th century on, there were restrictions on what they could wear (no gaudy clothes), the tonsure (or partially shaved head) they were obliged to wear, and the professions (e.g., butcher, barber, acting) and pastimes (e.g., hunting, gambling) they were forbidden to engage in. They were also legally protected from assault (to strike a cleric was to automatically incur excommunication) and from trial in secular courts.

In this period, especially in parts of England (e.g., Wales and parts of Norfolk) your priest may have been as good as married. Most priests kept housekeepers in the rectory, ideally women related to them. But unrelated women could become, in effect, common law wives. Such a woman was called colloquially in Latin a “focaria” or hearth-mate. Parishioners tended to turn a blind eye to this kind of shacking up just so long as priests left their own wives and daughters alone.

The priest of your "small village" was the leader of your community in spiritual matters, of course, but he was not the community leader since English parish life tended to be communally organized and self-governing (within the legal confines of manorial custom). But he was well known to everyone, better educated than everyone (though this might be just enough education to read through the Latin of the Mass and other services he had to perform), well-informed about events outside the parish via clerical networks, and he ranked among the more well-to-do members of your village because of his guaranteed income. By virtue of these qualities, his opinion was probably respected in important matters--unless, of course, he was an a-hole, which was not unheard of.

Sources: The various chapter introductions in Pastors and the Care of Souls in Medieval England (eds. W. Dohar & J. Shinners) give a well-rounded and fairly up-to-date account of medieval priests). Also useful: Peter Heath, English Parish Clergy on the Eve of the Reformation (1969), W. A. Pantin, The English Church in the Fourteenth Century (1955. rpt 1980), J.R.H. Moorman, Church Life in England in the Thirteenth Century (1955), Joseph Goering, “The Changing Face of the Village Parish: The Thirteenth Century,” in Pathways to Medieval Peasants ed. J. A. Raftis (1981), R.A. R. Hartridge, A History of Vicarages in the Middle Ages (1930, rpt. 1968), Katherine L. French. The People of the Parish: Community Life in a Late Medieval English Diocese (2001). EDIT: corrected sp of "altarage."