For a variety of reasons, Jewish religious law prohibits embalming and cremation. Embalming is the act of replacing all of the bodily fluids of a corpse with synthetic liquids that delay or arrest the body's decomposition. Embalming allows for open casket burial, and the ability to put the body on display for a long period of time. These practices go against Jewish cultural ideas of modest treatment of the body. One reason for the Jewish tradition of quick burial is that without embalming, the body will begin to deteriorate.
Cremation is the act of burning a corpse, under high heat, to remove all liquids and reduce the body to a small box of ashes and bone and tooth fragments. The rabbis who wrote the Talmud didn't like cremation because they believed that only a naturally decomposing body could be resurrected. Progressive Jews in the late 19th and early 20th centuries were interested in cremation as a more modern method of disposing of dead bodies, but knowledge of the Nazis' forced cremation of Jews during the Holocaust has made cremation unattractive to many Jews in the second half of the century.
Today, however, many people believe that cremation is more environmentally friendly than burial. Cremation is also less expensive in many places than a full burial, so deciding to cremate may fit better with Jewish ideas about keeping funerals as simple as possible, and accessibility of burial to all families, than a traditional burial does. Many traditional Jewish cemeteries will not provide space for cremains (the ashes of someone who has been cremated) to be buried. Some newer cemeteries will allow a family to bury cremains in the same casket or plot with a body. Some rabbis will officiate at funeral services before or after a cremation, even though cremation is not a traditional Jewish practice.
Even though embalming and cremation aren't Jewish practices, the Jewish relatives of an interfaith family may be just as likely to ask to be honored in these ways after they die as non-Jewish relatives are. It may be challenging to you if your religious practice or beliefs are different from the person making the last requests. It is a good idea to discuss what you want when you die with the people closest to you, and to be honest with each other about your feelings.
The Guide to Death and Mourning for Interfaith Families is also available as a downloadable PDF and Word document.