The process after the Pope dies

• When the pope dies, the dean of the Sacred
  College of Cardinals notifies the cardinals and calls a meeting —
  always held in the morning — that must begin no more than 20 days
  after the pope’s death. However, the cardinals already in Rome must
  wait 15 days for those who are absent to travel to the Vatican. Once at
  the Vatican, the cardinals must remain there and cannot communicate
  with anyone outside the area where the election is taking place. The
  maximum number of cardinal electors cannot exceed 120.

• The
  cardinals draw lots to select three members to collect ballots from the
  infirm, three "tellers" to count the votes and three others to review
  the results.
• Blank ballots are then prepared and distributed.
• After writing the name of one man on his ballot, each of the active cardinals
  — those under 80 years of age — walks to an altar and pledges to
  perform his duty with integrity. He then places his ballot in a
  container, which is covered by a plate.
• After all votes are cast, the tellers tally the ballots and the result is
  read to the cardinals. If a cardinal receives two-thirds plus one of
  the vote, he is the new pontiff.
• If there is no winner, another vote is taken. If there is still no winner, two more votes are scheduled for the afternoon.
• After
  the votes are counted each time, the ballots are burned. If there has
  been no winner, a chemical is mixed with the ballots to produce black
  smoke when they are burned. Sight of the black smoke emerging from the
  roof of the Vatican Palace tells those waiting in St. Peter’s Square
  that a pope has not yet been selected. When a winner has been selected,
  the ballots are burned alone, and the white smoke indicates there is a
  new pope.
• Traditionally, the winner had to
  garner two-thirds of the vote plus one, but John Paul II changed that
  in 1996. He ruled that if, after 12 or 13 days there is still no
  winner, the conclave could invoke a rule — by majority vote — that
  would permit the selection of the pope by an absolute majority.
• Once
  there is a winner, the pope-elect is asked if he accepts the decision.
  (Pope John Paul II reportedly accepted his election with tears in his
  eyes.) If he does, the dean asks what name he chooses and announces it
  to the cardinals, who then come forward to offer congratulations.
• The
  oldest cardinal then steps out on a balcony overlooking St. Peter’s
  Square and says to the crowd, "Habemus papam" — "We have a pope." He
  then introduces the pope, who steps out on the balcony to bless Rome
  and the world.
• Many popes have been formally
  installed with a coronation, but Pope John Paul II refused a coronation
  and was installed as the pope during a Mass in St. Peter’s Square.

The selection processes are very old processes (and the most common of these was, interestingly, discussed in Dan Brown’s book, Angels and Demons).  There are two other methods of electing a Pope, namely a unanimous voice vote or a unanimous vote of a committee of cardinals.  There is a pretty interesting history of the selection process in CNN’s Special Report.  The BBC also has a fairly interesting article about the selection process which also includes the destruction of the Pope’s Fisherman’s Ring and papal seal as well as the Pope’s funeral and burial.

The BBC has a list of possible successors to Pope John Paul II.  It is interesting to see a Black African candidate although it has been said that the Church may not be quite so progressive as to elect a Black Pope this time around.

The official web site for the Holy See is here.