Answer:
   
   [Note that this description is derived mostly from the Zohar, a 12th
   century work that evolved from the mystical writings of R' Shim'on bar
   Yochai (2nd century). The Zohar is the central writing of Kabbalah.]
   
   In general, and in contrast to some other religions, in day-to-day
   life Jews don't pay much attention to questions like this. The focus
   of Jewish life is living according to G-d's will as expressed in the
   Torah. What happens afterwards is up to G-d.
   
   That said, traditional Judaism does address this question. To put it
   shortly, our beliefs in resurection and afterlife vary widely. Some
   believe it is part of the Messianic era. Some consider it an era of
   its own, after the messianic one. It's a matter of debate in Jewish
   tradition as to whether the post resurection life is permanent, or
   temporary. Nachamides believes that the ultimate reward, the "World to
   Come" is that post-resurrection life, and therefore it must be
   eternal. Maimonides opines that the ultimate reward is the relatively
   direct experience of G-d that a soul can have when not encumbered with
   a body and its desires. Therefore he understands the phrase "World to
   Come" to refer to the non-physical existance after life, and that's
   man ultimate reward. He returns to that reward after a second,
   resurrected life. This is because Maimonides believes it's because man
   can only face his judgement in the same condition as when he sinned.
   Since he sinned while in a body, he is returned to that body to be
   judged. R' Yosef Albo agrees with Maimonides that the post
   resurrection life isn't permanent. To be specific, he believes that
   the lifespan will be 1,000 years -- the length of time Adam would have
   lived after eating from the forbidden fruit (had he not given away 70
   years for someone else). His reason for this second life, though, is
   very different. Albo writes in the Ikkarim (Fundamentals) that in this
   life, man masters the art of self-perfection in the face of adversity
   -- disease, threat of poverty, and everything else that could go wrong
   in life. In the next life, the only challenges are internal, there
   will be no external impediments. It's therefore a second step in
   personal development, allowing for more refinement in one's ability to
   enjoy the World to Come upon return. In the early 20th century, Rabbi
   Abraham Isaac Kook (cheif Rabbi of British Mandate Palestine) wrote
   that the resurection occurs by the end of the illusion that separates
   this universe from heaven.
   
   Now, for the more mystical explanation. Keep in mind that there are
   dissenting viewpoints, though this is the dominant one.
   
   A living person consists of both body and soul. Both are complex in
   structure and this short answer can't possibly address the details. To
   summarize briefly, when the body dies, if the person merits it, a
   small portion of the soul remains with it to keep it connected with
   the soul's source, anticipating the general revival of the dead at the
   time that G-d decrees. Different parts of the remainder of the soul
   may go to different places. One might be reincarnated into a new body
   in an attempt to rectify another of its spiritual aspects, or for
   other purposes. One part might go to a level of Paradise. Another
   might go to Gehinnom for a period, to remove the sins of that life and
   prepare it for a future one. Another part might join temporarily with
   an already living person, to assist it with its rectification and in
   the process gather more merit. The reassignments of the soul continues
   until the time that G-d decrees.
   
   Rabbinic afterlife teachings varied in different places and times, and
   was never synthesized into one coherent philosophy. As such, the
   different descriptions of the afterlife are not always consistent with
   each other. This is especially true for the descriptions of "Olam
   Haba", the world to come. In some rabbinic works this phrase refers to
   the messianic era, a physical realm right here on Earth. However, in
   other works this phrase means Gan Edan, Paradise (in Heaven, so to
   speak), a purely spiritual realm. At various points in the afterlife
   journey, the soul is said to encounter:
     * Hibbut ha-kever, the pains of the grave
     * Dumah, the angel of silence
     * The angel of death
     * The Kaf ha-Kela, the catapult of the soul
     * Gehenna (purgatory) and Gan Eden (Heaven; Paradise)
       
   A discussion of the classic rabbinic view of the afterlife, including
   these topics and more, can be found in an essay by Rabbi Zalman
   Schacter Shalomi called "Life in the hereafter: A tour of what's to
   come", found at [5]http://www.elevated.fsnet.co.uk/index-page13.html.
   
   Gehenna is fairly well defined in rabbinic literature. It is sometimes
   translated as "hell", but Jews must take note that the Christian
   version of hell is different from the Jewish view of Gehenna. Some
   Christians believe that hell is an abode of eternal torment where
   sinners go, and is also for anyone who does not accept Jesus as their
   messiah and G-d. Other Christians believe Hell is a place of
   separation from G-d (which, for Christians, is torture enough), from
   which believers are eventually saved by Jesus. Roman Catholics believe
   that Hell is a place of eternal suffering--physical, mental and
   spiritual suffering. In the Roman Catholic view of Hell, the physical
   pain is constant and severe; but the worst torture of Hell is the
   knowledge that they will never see G-d and that they will remain in
   Hell for eternity. For Roman Catholics, Hell is permanent and eternal.
   For Roman Catholics, the soul that has deliberately and knowingly
   disobeyed G-d's commandments in life and that remains in a state of
   mortal sin upon death has through it's own free will damned itself to
   Hell for all eternity. Roman Catholics also have the notion of
   Purgatory, which is for souls that are truly repentant, but not in the
   state of grace upon death. Purgatory is similar to Hell in that there
   is physical suffering, the Roman Catholic belief is that the soul will
   return to G-d when it is purged of its sins. Purgatory can last a day
   or thousands of years depending on the amount of purging the
   individual soul requires.
   
   However, for Jews, gehenna--while certainly a terribly unpleasant
   place--is not hell. The majority of rabbinic thought maintains that
   people are not tortured in hell forever; the longest that one can be
   there is said to be 12 months. It is a spiritual forge where the soul
   is purified for its eventual ascent to Gan Eden [Heaven], and where
   all imperfections are purged. [In this sense, it is somewhat similar
   to the Roman Catholic purgatory, however the time period has a
   definate maximum]. Gehennom (lit: the valley of Hinnom, in Jerusalem;
   i.e. hell) is the sinner's experience in the afterlife. In other
   words, it's the same "place" as gan eiden (lit: the garden of Eden;
   i.e. heaven) -- it's the perspective of the individual that makes it
   one or the other.
   
   In some descriptions of the afterlife, we find that beyond Gan Eden
   there is a little known realm called the otzar, the divine treasury of
   souls; this is also called the tzror ha-hayyim, the bundle of life.
   This otzar is a transcendent realm of human souls, in the highest
   spheres of creation. Before souls are born they are said to come from
   this treasury, and they return they at some point after death.
   
   Souls are said to originate in a realm called the 'guf' (Avodah Zarah
   5a, Nedarim 13b, Yevamot 62a), from which they descend to the earthly
   real to animate human bodies. After death, these souls return to the
   otzar, or tzror ha-hayyim. (Shabbat 152a; Pesikta Rabbati 2:3)
   
   According to the Kabbalah [Jewish mysticism] every human has at least
   one element in their soul; with the proper study a person can
   eventually develop two higher levels of the soul. A common way of
   explaining the three parts of the soul is as follows:
    1. Nefesh - the lower part, or animal part, of the soul. Is linked to
       instincts and bodily cravings.
    2. Ruach - the middle soul, the spirit. It contains the moral virtues
       and the ability to distinguish between good and evil.
    3. Neshamah - the higher soul, or super-soul. This separates man from
       all other lifeforms. It is related to the intellect, and allows
       man to enjoy and benefit from the afterlife. This part of the soul
       is provided both to Jew and non-Jew alike at birth. It allows one
       to have some awareness of the existence and presence of G-d.
       The "Raaya Meheimna," a later addition to the Zohar, posits that
       there are in fact two more parts of the human soul, the chayyah
       and yehidah. These parts were considered to represent the
       sublimest levels of intuitive cognition, and were only within the
       grasp of very few individuals.
    4. Chayyah - The part of the soul that allows one to have an
       awareness of the divine life force itself.
    5. Yehidad - the highest plane of the soul, in which one can achieve
       as full a union with G-d as is possible.
       
   According to the Zohar, after death each aspect of the soul undergoes
   a different experience on the afterlife journey. The lower levels of
   the soul are purified and purged of physical and emotional
   attachments, while the higher levels experience transcendental bliss.
   The nefesh temporarily remains with the body in the grave, undergoing
   the Hibbut Ha-Kever, the suffering of the grave. Simultaneously, the
   Ruach experiences Gehenna for 12 months. "Gehenna is conceived of as a
   purification process in which the psychic remnants from the previous
   life are purged and transformed. This purgation process lasts only
   twelve months and is tormentingly painful in direct proportion to each
   individual's lived life experience. [Simcha Paull-Rapahel ] After
   leaving Gehenna, the ruah then permanently enters the Lower Gan Eden.
   
   After death the Neshama, since it not subject to being tainted by sin,
   goes to Gan Eden Elyon, the Upper Gan Eden, where it experiences
   divine reward and bliss. The hayyah and yehidah also return to Upper
   Gan Eden immediately after death, and become as one with G-d as is
   possible. "Those who have awakened these dimensions of their being are
   able to perceive the infinite grandeur of the divine realms, to enter
   the everflowing celestial stream - described by the Zoha as the
   "bundle of life". [Paull-Rapahel]
   
   Given all this, what happens to the soul of the nonbeliever? The most
   common belief in contemporary traditional Jewish communities is that
   all souls go to the after-life. Nearly all, barring a handful or two
   in all of human history, eventually end up in Gan Eden (roughly:
   heaven), even non-believers. Maimonides (a medieval Jewish thinker)
   opined that non-believers cease to exist upon death. His reasoning was
   that the ability to exist eternally is G-d's, and is only acquired by
   the soul to the extent that the soul knows of, and therefore shares
   some of the form of, G-d. This opinion was more popular in the midevil
   period, but no longer captures much attention, since around the early
   19th century. At that time, the Chassidic and Mussar movements
   influenced Orthodox thought. The Aristotilian influence of the
   medieval thinkers like Maimonides faded in favor of other, equally
   old, approaches to the problem. All of these notions have roots in the
   Talmud (our earliest written rabbinic texts) and earlier. It is just a
   matter of which approach to G-d from within that tradition people
   follow. [Note: While you may have heard of Chassidim, there are few if
   any Mussarists left post-WWII. It was an Orthodox movement based on
   personality improvement and stressed the inter-personal commandments.]
   
   As for the question of Purgatory. Again, there is no one Jewish
   position on the subject, even if we limit ourselves to the traditional
   Orthodox position. The Talmud refers to the deceased going to a Word
   of Truth or going to the heavens--without distinction. It is generally
   assumed these are homonyms, but these quotes still speak of a single
   afterlife. Others speak of the Garden of Eden and Gehenna. Neither
   could be meant literally, as Adam was in the literal Garden before
   death, and the valley of Hinnom (Gei Hinnom, in Hebrew) is a valley in
   current Jerusalem (where the Canaanite locals practiced human
   sacrifice by passing their children through the fires for Molech).
   Some therefore understand this to mean that Eden vs Gehenna is not a
   difference in "location" but rather in how one experiences the
   afterlife. Someone who spent life developing an appreciation for G-d
   and Truth will find it as pleasant (Heb: eden) as the garden, those
   who developed interest in other pursuits will find the experience
   hellish. A number of sources, such as R' Chaim of Vilozhin (founder of
   the current Yeshiva movement, late 18th early 19th cent) and R' Israel
   of Salant (founder of the Mussar movement, late 18th cent), describe
   the fires of gehenna as those of shame. Facing the truth of what one
   could have been and seeing what one was. The number of people subject
   to a permanent "stay" in gehenna is very small. The Talmud (Tr
   Sanhedrin, 11th ch) names 4 people up to their day who qualified.
   Otherwise, the experience itself is atoning, creating a person who is
   capable of enjoying the presence of G-d. For these few people, they so
   identified themselves with sin that to abandon sin would be to lose
   their essence.