Spiritism is a loose corpus of religious faiths having in common the general belief in the survival of a spirit after death. In a stricter sense, it is the religion, beliefs and practices of the people affiliated to the International Spiritist Union, based on the works of Allan Kardec and others. Formed in France in the 19th century, it soon spread to other countries, but today the only country where it has a significant number of adherents is Brazil.
The fundamental principles of Spiritism, enunciated by Allan Kardec in his seminal work The Spirits Book, are: (i) A belief in the existence of spirits - non-physical beings that live in the invisible or spirit world - and (ii) the possibility of communication between these spirits and living people through mediumship. There is a clear difference between the terms "Spiritism" and "Spiritualism": Although there are many similarities between the two, they differ in some fundamental aspects, particularly regarding man's quest toward spiritual perfection and the manner by which the followers of each practice their beliefs.
Spiritism teaches reincarnation or rebirth into human life after death. This basically distinguishes Spiritism from Spiritualism. According to the Spiritist doctrine, reincarnation explains the moral and intellectual differences among men. It also provides the path to man's moral and intellectual perfection by amending for his mistakes and increasing his knowledge in successive lives. For this reason Spiritism does not accept rebirth in animals as this would be retrogressive.
Source: Wikipedia, "Kardecism", available under the CC-BY-SA License.