You are posing a question which medievalists themselves can’t answer with certainty. There are two schools of thought. One, long the orthodox position until the later decades of the 20th century, holds that medieval people, especially the peasantry, were scarcely Christianized at all. Churchmen simply lacked the resources and training to effectively teach Christian doctrine to the masses in all but the most piecemeal way. Even long into the Reformation and the spread of printing, large pockets of Europeans remained backward, ill-informed, and frankly grossly superstitious about their faith. (Jean Delameaus’s 1971 Le Catholicisme entre Luther et Voltaire, trans. Catholicism Between Luther and Voltaire: New View of the Counter-Reformation, is a classic exploration of this position.) The second school, which nowadays seems to hold the upper hand, argues that, however rude and crude laypeople might be about the details, they embraced the rituals and basic beliefs of Christianity and used them in their daily lives. (It’s not exactly the pioneering statement of this school of thought, but Eamon Duffy’s magisterial 1992 The Stripping of the Altars: Traditional Religion in England, 1400–1580 is one of the most exhaustive explorations of this idea. There are many recent local studies supporting this approach.) By the way, it’s worth noting that “peasants” is a very unwieldy term to characterize a very diverse group of people in various places at various times. There just never was such a thing as a “medieval peasant” through whom we can make generalizations.

But even setting aside this debate, as with many questions about the lives of medieval peasants, there are too many variables intruding on your question that make generalizations impossible. So an answer to your question about a medieval peasant’s grasp of Christian dogma would have to first ask, when does your peasant live? The religious instruction available to a peasant in the 8th century before the parish system had effectively been established in (western) Europe in by the 11th century would be different from one living in the 14th. Where does your peasant live? Someone near a town or monastery would be probably have access to better avenues of religious instruction than someone in more remote areas. Peasants living in the early 13th century and beyond would likely encounter itinerant preachers from the new mendicant orders (e.g., Franciscans and, especially, Dominicans) who could offer them sounder religious instruction than the average parish priest. Chronology would again affect what a peasant was expected to know about Christianity. The definition of the seven sacraments was not nailed down until the 12th century. Confession especially underwent a transformation from something that measured sinful acts to something that weighed sinful intentions as well, a matter more for private examination than public infamy (though I’m oversimplifying here). The understanding of the Eucharist likewise evolved over the course of the Middle Ages. Again, by the 12th century the idea that the eucharistic bread and wine became the Real Presence of Jesus Christ at the consecration during the Mass became a fixed dogmatic idea. By the 12th century, this reverential understanding of the Eucharist meant that parishioners received only the bread at Mass, not the wine too. This markedly affected how ordinary Christians perceived and reverenced the Eucharist (see Miri Rubin’s 1991 Corpus Christi: The Eucharist in Late Medieval Culture).

Stepping back some, we can broadly say that, all in all, there were minimal expectations about what peasants—or any other lay person—was expected to know about the Christian faith. From early on, lay Christians were expected to know the words (in Latin and, ideally, their vernacular) of basic Christian prayers: the “Our Father,” the “Hail Mary,” and the “Creed” and how to make the sign of the cross. The Creed, traditionally divided into 12 sections, one for each Apostle, encapsulated the core dogma of Christianity: God as a trinity, Jesus as God incarnated through the Holy Spirit and Mary, who then was crucified, died, and rose from the dead, the authority of the Church, etc. The very important Fourth Lateran Council of 1215, an effort to reform and professionalize the clergy, mandated that every Christian was obliged to make a confession and receive the Eucharist at least once a year. Just once, usually at Easter. Local variation allowed more frequent confession and reception but realistically ordinary Christians would receive either sacrament at most 3 or 4 times a year. All Christians would be baptized as infants (and parents and godparents were obliged to know the verbal formula for baptism in case a priest was absent in an emergency). All Christians ideally would be confirmed at some point in their childhood (churchmen were vague about exactly when) but it’s unclear whether most Christians were at least in the earlier Middle Ages since only a bishop could administer this sacrament. Marriage was defined as a sacrament in the 12th century. Ideally, a priest should be present to bless a marriage, but the sacramentally binding oath was exchanged by the couple, who could marry one another without the presence of anyone, even a priest. Dying Christians ideally received “Extreme Unction” or the “Last Anointing” but, again, it’s unclear how many did. Canon lawyers and theologians determined that even parish priests were obliged to know only “a little more” about Christianity than their parishioners.

The complete bible, which became the backbone of the Reformation, well, at least in theory, was not available to ordinary Christians or even ordinary parish priests in the Middle Ages, though large chunks of it, especially its prominent stories (Adam & Eve, Noah, Moses, New Testament stories, etc.), would be explained in sermons.

I could turn your question back on you and ask how well-versed in the dogma of their various denominations are modern Christians? The answer would be, of course, not very. Just as in the Middle Ages, dogma is really less important to ordinary believers than the private consolation and public performance of their faith. People grasp basics but are fuzzy on details. Just ask a modern Christian, for instance, to explain the Incarnation, the trinity, the Catholic idea of Real Presence. I would suggest that people get focused on dogma only when it serves their personal or public agendas: pro-social justice, anti-abortion, anti-capital punishment, anti-gay etc.

Sources: Bernard Hamilton’s classic Religion in the Medieval West (1986) is a great short introduction. R. N. Swanson’s Religion and Devotion in Europe, c. 1215-c. 1515 (1995) is a more sustained survey. Daniel Bornstein has edited a nice volume touching many facets of this question in A People’s History of Christianity: Medieval Christianity (2010), Medieval Christianity in Practice, edited by Miri Rubin (2009) is a valuable close focus on practice, though strangely it has almost nothing to say about veneration of Mary, a cornerstone of medieval belief. James Halverson edited Contesting Christianity: Readings in Medieval Religion and Culture (2008), which excerpts some of the groundbreaking articles and books on the subject.

EDIT: Changed a preposition.