Supersessionism, in short, is the belief that Christianity supersedes Judaism.
Obviously, any religious person is going to assume that her own religion is the "best" one in some sense. In this, Christianity is not unique: if a member of any of the world's religions, Christian or otherwise, were to come to the conclusion that a different religion were superior, she would convert to it if she could. But Christianity has an unusual family relationship with Judaism that it does not share with, say, Buddhism or Shinto, and this relationship colours Christianity's relationship with its mother religion in a profound way.
In order to understand what it means to be a supersessionist, one must first understand the concept of covenant. Classical Judaism maintains that God made a contract with his people -- that's what "covenant" means (and "testament" too, but we'll get to that in a minute). Stories about the contract, and the content of the laws themselves, are contained within the Bible, which in Hebrew is called the Tanakh.
The laws themselves are famously detailed. They include restrictions on what Jews may eat and wear, and they offer lengthy lists of punishments for a vast range of crimes -- many of which have no relevance to an affluent Western reader today, even as some topics that are of pressing concern to modern readers remain unaddressed. Some Jews try their best to follow all these laws, believing they are still in force in exactly the form they were written; others try to update and reinterpret them in various ways; still others don't concern themselves with the law at all, finding their social and spiritual strength in other places.
Jesus drew most of his authority on the Tanakh: he quotes it frequently, he is addressed as rabbi, and his followers clearly believed that he was the messiah whose coming was prophesied in texts like Isaiah and Samuel. Such a position would have made no sense to non-Jews (and indeed it didn't, as Roman persecution of Christians would later prove). At the same time, however, Jesus taught that purity law was not a part of God's plan for his people, and he encouraged his followers to eat whatever they wanted with whomever they wanted. He even encouraged them to drink his blood, a command that is not just strange but expressly forbidden by Jewish law.
So should Christians follow the Jewish law or not? Should they even read it? Is it relevant to Christianity at all? Some early Christian sects tried to separate Jesus from the Jewish covenant entirely, but they were unsuccessful for various reasons. Paul of Tarsus, for his part, argued passionately in favour of abandoning Jewish dietary restrictions, circumcision, and Sabbath observances. Paul argued that Jesus' death and resurrection brought about a new covenant -- that is to say, a new contract, a new testament. All that stuff in the Jewish Bible is just the old testament, a contract that Jesus tore up and rewrote.
For better or for worse, Christians today follow Paul's lead: they do not keep kosher and they do not worry too much about what Leviticus says. The path between Jews and Christians has forked so dramatically that many conservative churchgoers don't even realize that their "Old Testament" is the Jewish Bible. If they have any awareness of the issue at all, they'll say that the Israelites were punished for rejecting God, and that laws against eating pork and whatnot were simply a way to keep a rebellious people in line until the arrival of Christ. Anyway, "the Jews" killed Christ, and any favour they may have had with God beforehand has clearly been annulled when "they" murdered His Only-Begotten Son.
Needless to say, supersessionist assumptions can lead to a particularly vicious strain of anti-Semitism. Not only is Judaism the "wrong" religion (narrow-minded Christians might say that of any religion, including other sects of Christianity), but it has a special kind of wrongness not shared by Zoroastrians or Jains. Judaism is an outdated religion, whose numerous picky and useless laws were intended for a misguided people of another time and place. Anyone who continues to keep kosher must be doing it out of sheer ignorance, perversity, or unwillingness to see the obvious truth.
This attitude toward Judaism appears very early in Christian literature, and it survives, in a somewhat modified form, in the dispensationalist assumptions favoured by many evangelical Christians today. As early as the second century, Justin Martyr could tell his Jewish friend Trypho:
What need, then, have I of circumcision, who have been witnessed to by God? What need have I of that other baptism, who have been baptized with the Holy Ghost? ... These words have neither been prepared by me, nor embellished by the art of man; but David sung them, Isaiah preached them, Zechariah proclaimed them, and Moses wrote them. Are you acquainted with them, Trypho? They are contained in your Scriptures, or rather not yours, but ours. (Dialogue with Trypho 29, emphasis mine.)
Liberal Christians tend to be embarrassed by supersessionism, and they have come up with numerous strategies to avoid it. They talk, not of the "Old" and "New" Testaments, but rather of the "First" and "Second" Testaments. This shift in vocabulary is meant to demonstrate that both contracts are still perfectly valid; they simply speak of God's relationship to his creation in two different and complementary ways. Liberal Christians also resist the idea that "the Jews killed Christ," explaining that historically, crucifixion was a Roman punishment, not a Jewish one, and besides, the point of the crucifixion story is that everyone who loves Jesus betrays him whether they mean to or not. It's not the Jews who hurt Jesus, not as such; it is his friends, his students, his travelling companions.
Roman Catholicism, too, has been moving away from the "Jews killed Christ" model of the Middle Ages. In the thirteenth century, the Fourth Lateran Council had some pretty foul things to say about Jews, but the most recent edition of the Catechism is much more tolerant and generous:
When she delves into her own mystery, the Church, the People of God in the New Covenant, discovers her link with the Jewish People, "the first to hear the Word of God." The Jewish faith, unlike other non-Christian religions, is already a response to God's revelation in the Old Covenant. To the Jews "belong the sonship, the glory, the covenants, the giving of the law, the worship, and the promises; to them belong the patriarchs, and of their race, according to the flesh, is the Christ"; "for the gifts and the call of God are irrevocable." (Catechism of the Catholic Church, second edition, section 839.)
Nevertheless, plenty of Catholics don't read the Catechism, and even Christians of a liberal stripe sometimes harbour supersessionist assumptions without realizing it. Anyone who says, even in passing, that the "Old Testament god" is "vindictive" and that the "New Testament god" is "forgiving" is talking like a Marcionite. In a famous Internet meme, a bunch of Jewish laws are paraded in front of Dr. Laura Schlessinger in an effort to demonstrate that they are petty, pointless, and outdated, which would come as news to the Orthodox Jews who take those laws seriously. And Justin's claim that the Bible "belongs" to Christians -- that the Jews don't even know how to read it properly -- is taken as self-evident by many well-meaning Christians. "But of course Isaiah 53 is talking about Jesus. Who else could it be talking about?"
Is it possible to be a committed Christian without hating Jews? Can one take Jesus' claims of messiahship seriously even as they abandon the Law that he himself quoted approvingly? Paul seemed to think so, but Paul has been accused of being the first anti-Semite (and a prude and a misogynist to boot). Christianity's origins as a branch of Judaism has led to a lot of tensions between the two religions, as their members argue over the proper way to interpret prophecy.