"As an invention it is very old. It was in use in the days of Pythagoras, about 540 B.C. According to a French historical account of the philosopher's life, his sect held frequent séances or circles at which a mystic table, moving on wheels, moved towards signs, which the philosopher and his pupil, Philolaus, interpreted to the audience as being revelations supposedly from the unseen world." -Encyclopedia of Psychic Science.

The first recorded use of a planchette seems to have been Chinese, used to spell out characters in sand or incense ashes. A planchette resembling a modern divination "wand" or "dousing" rod was simply referred to as chi, While trance channels were known as "wu", (Chi is a Chinese word meaning aliveness, life force energy or life breath. - also known as Ki, Qi or Prana. The Definition of Chi: "Theories of traditional Chinese medicine assert that the body has natural patterns of qi associated with it that circulate". Wang Ch'ung in the first century CE wrote that "among men the dead speak through living persons whom they throw into a trance; and the wu, thrumming their black chords, call down the souls of the dead, which then speak through the mouths of the wu" (quoted in Klimo 80-81).

Fuji (Chinese: 扶乩/扶箕; pinyin: fújī; Wade–Giles: fu chi) is a method of "planchette writing", or "spirit writing", that uses a suspended sieve or tray guiding a stick, for writing in sand or incense ashes. "Fu chi" Began around the Ming Dynasty (1368-1644 CE). In the magical practice of late and post-classical periods an instrument is sometimes provided through which the ghost speaks. The ghost is summoned into a corpse, either that of the victim of sacrifice or one selected, as in the scene in Lucan's Pharsalia, from the graveyard in which the incantation takes place. The papyri give directions for calling the spirit into the corpse, and coercing it to reveal the future. (Halliday 242-44).

Whether or not a board was used, the endeavor to communicate with spirits is nothing new. Sacrifices made of unborn males were recorded in accounts of M. Tullius Cicero, Against Vatinius. The belief was that a person, newly deceased, would be in between the world of the living and dead, and thus able to breach the gap, and communicate. Specifically, the aim was to know the future, rather than speaking to a lost relative. In the magical practice of late and post-classical periods an object as an instrument is sometimes provided through which the ghost speaks. The muses were also channeled spirits: the muse-inspired poet or singer was thought to be the mere bodily vehicle for the singing of the muse ("Sing in me, muse," begins Homer's Odyssey).

The Greek oracles at Dodona and Delphi and other sites were trance-channelers who would prophesy by sinking into a trance and being possessed by discarnate spirits--some of the famous ones by a single spirit, or what we would today call a "spirit-guide."

EDIT: I forgot to reference something in answer to your question about the beginning of the spiritualism movement of the 1900's. It is a coincidence that I just read a book about the Fox sisters. Spiritualists often set the date of March 31, 1848 as the beginning of their movement. On that date, Kate and Margaret Fox, of Hydesville, New York, reported that they had made contact with a spirit. It is thought by many that it all began with deception, a game that got out of hand and led to a widening interest in, and played an important role in the formulation of spiritualism.

The Fox sisters utilized a method of knocking (and one sister was allegedly able to crack her joints loudly, while the other used a set of blocks between her knees, which she clapped together). The number of knocks began with sets of one, meaning "No", two meaning "Yes". Many people were compelled to visit them, to hear these knocks. "They", or rather the knocks, alleged to be a murder victim buried in the Fox home's basement, accused a man of murder.

The Fox sisters attracted much attention, and the tales spread. Eventually an associate of the sisters assigned an alphabet to the number of knocks, so that the "spirits" could "speak" through them. After moving from their "haunted" home, and having the knocks follow them, it was widely believed that the Fox sisters could communicate with the dead. In 1888 Margaret and Kate confessed, however Margaret tried to recant her confession a year later, upon the ruination of her family's reputation, it was to no avail.

Whether or not they made contact with anyone (deceased) through these knocks, is open to conjecture. I point to it because it is in reference to the beginning of the spiritualist movement, but also because of the similarities in the means of "communication" to that of the "Ouija" board. When asked a question, the answer would be "rapped out" and recorded, letter by letter, by someone hand-writing the messages.

The social climate was ripe for the mania of speaking with the dead. There are many reasons, such as the high mortality rate, especially in children. There were more unknowns than answers for the grieving Victorians, for whom death was accentuated with rigid adherence of etiquette and religion as an ever-present fact of life. There was a pervasive inability to control elements such as access to the limited healthcare available, and a widespread fear of disease; not an irrational fear, considering the times of plague within memory of the living. Distrust of vaccination was at a peak, (there were anti-vaccination movements, with protests against mandatory vaccination, more than today). This, in an age of ignorance and need that was a culture to feed beliefs in search of solace. They wanted, at least to know their loved-ones were safe in heaven.

This was really interesting to research, thanks OP!

I see someone else has tackled the first half of the question, and so I think I will take a crack at the second. Spiritualism might be tied to a broader cultural trend in the Victorian era called the "crisis of faith." This has often been presented as the notion that, under fire by expanding secular science, traditional Christian beliefs started to lose followers. Geologists were demonstrating that the Earth was much, much older than previously believed, calling the Biblical narrative into question. Dinosaur mania swept the English-speaking world, and the age and size of the fossils contributed to this staggering sense of "deep time," of which human civilization forms only a fragment. Most importantly, Charles Darwin and others shook society when they declared that humans are directly related to other animals and were formed not by a divine hand but by natural processes. (It is worth noting, perhaps, that Darwin himself writes of the first life as being "originally breathed by the Creator into a few forms or into one"--however, the implications of his work were immediately felt, as the Huxley-Wilberforce debate of 1860 demonstrates. It is also worth noting that Darwin's work was not as unexpected a shock as some have suggested: there were previous theories about "the natural history of creation.") This does not take into account the very lively and interesting intellectual processes within established religious communities that engaged with these proceedings. I could expand on this if anyone is curious, but for now I will close on "In Memoriam A. H. H." by Tennyson, which dramatizes this process of doubt. Tennyson references fossils and their mute evidence of the way time destroys what comes before: "From scarped cliff and quarried stone / She [Nature] cries: "A thousand types [species] are gone: / I care for nothing, all shall go." (Taken from Canto LVI, the canto that also gave us the phrase "Nature, red in tooth and claw.") At the end of the poem, Tennyson's speaker writes of finding fragile hope for the afterlife, in which he will see his friend again.

This crisis of doubt led to powerful religious revivals, such as the third Great Awakening sweeping the United States. These movements often emphasized the ability of the individual to make a direct connection with God, bypassing traditional sources of spiritual authority. They were led by charismatic individuals who claimed to have a personal connection to supernatural forces. I will not dwell on this here, but we see parallels in spiritualism. I will use Theosophy as an example, one of the most organized and widespread spiritualist movements of the time. Helena Blavatsky, a Russian spiritualist, combined elements from world religions and Western esotericism to create an "occult science." This formed, in many ways, an alternative, sheathed in pseudoscience, to mainstream science. It taught an alternative history accessed through clairvoyance and argued that humans, after all, are not merely animals but the descendants of incorporeal beings. This was, in a way, more palatable to some than what science was offering. This time also saw the massive popularity of spiritualism as entertainment. Huge audiences crowded into halls to see mediums do amazing feats: producing flowers from thin air, moving things with their minds, levitating, even manifesting ghostly apparitions and ectoplasm. Seances and mediums enjoyed unprecedented success. By early 20th century, many of these processes had been thoroughly debunked.

It must be stressed, I think, that this was a time that the first real understanding of the unconscious and the workings of the human mind started being uncovered. Scientists demonstrated ways in which electricity and chemistry could produce action in tissue (Duchenne de Boulogne went far beyond what Luigi Galvani's electrical experiments to make muscles move by inducing involuntary expressions in human faces): could this be a step away from discovering the vital essence controlling the human body? It was, perhaps, not such a leap to imagine that these advances in science would lead to a quantifiable interaction with some form of "great beyond," that humans would (to use the often-repeated metaphor) lift the veil between this world and the next.

Further reading:
Leonard, Todd. Talking to the Other Side.
Weisberg, Barbara. Talking to the Dead.

Though of a slightly later era:
Sir Arthur Conan Doyle's strong belief in the supernatural: http://www.victorianweb.org/authors/doyle/spiritualism.html
An look into an alleged medium and the man who debunked her, Harry Houdini: http://www.historynet.com/mina-crandon-harry-houdini-the-medium-and-the-magician.htm