Answer:
   
   This question refers to the disputation between the Ramban
   (Nachmanides, not the Rambam) and Pablo Christiani, a Jew from
   Provence who became a Karaite and later a convert to Catholicism.
   Christiani had King James of Aragon's ear, and was able to convince
   the king to host the dispute. It was written up in a polemic by R'
   Chaim (Charles) Chavel, for which there is an English translation
   published by Shilo (1983), titled "Disputation at Barcelona". It is
   also the topic of Rabbi Dr. David Berger's PhD thesis and much of his
   book "Jewish-Christain Debate in the High Middle Ages", published by
   Aaronson in 1979. Some historical context is provided in from
   "Barcelona and Beyond: The Disputation of 1263 and Its Aftermath" by
   Robert Chazan, Berkeley: University of California Press (1992).
   Lastly, "The Disputation", a play re-enacting the debate, is available
   on video from a number of anti-missionary groups. (Try
   [5]http://www.outreachjudaism.org/)
   
   First, the use of "the" is probably incorrect, as there were many such
   disputations. They started around 1240 CE, when Nicholas Donin
   (another convert) challenged the local Rabbis to defend the Talmud
   against challenges of racism and anti-Christianity in the Talmud. They
   won the battle but lost the war, they won the dispute but the Talmuds
   were burned by the cartload anyway.
   
   Another famous disputation ran for two years. The Tortosa
   "disputation" (1413-14), between St. Vincent Ferrer and the apostate
   Geronimo and the local Jewish leaders. In this disputation, they let
   the Jews defend the gemara, but not attack Christianity. When the
   French Jewish community fled in the 1300s, and the Spanish and
   Portugese were force out in the 1490s, the trend of disputations
   ended.
   
   The dispute at Barcelona was near unique in that it let both sides
   have equal say. The key topics were Christianity, the Jewish
   definition of messiah, and proving the messiah hadn't yet come.
   Nachmanides relied heavily on logic, rather than purely citing
   sources. Again, Nachmanides won the battle, but lost the war--he had
   to flee Spain. The issues discussed in the debate included:
     * Whether Isaiah 53's suffering servant described Jesus.
     * Whether the Talmud, when it speaks of the messiah being alive in
       its day, meant that the messiah had come. In response to this,
       Nachmanides distinguished between the messiah being born, and the
       messiah coming. Moses didn't come until the declaration "Let my
       people go!" and clearly nothing parallel has happened to start the
       messianic era.
       
   An interesting side impact of this disputation was the result of the
   Ramban's willingness to use the Christian chapter system for citing
   verses in the dispute. This played a large role in the acceptance of
   the system amongst Jews, and its use today.
   
   Note that Pablo Chistianity trained under the Dominicans, the people
   behind the Inquisition. So the connection between the disputation and
   the later expulsion is probably significant. Also, when looking for
   books on Nachmanides in the Library of Congress, look under P for
   "Bonatruc ca Porta", his Castillian name.