Different sources give the literal meaning of the Hebrew word "geniza" or "genizah" as "container" or "to bury or hide." However, it is generally used for a synagogue depository where worn-out texts are stored, "a kind of book tomb" as Matthew Battles put it in his book Library. The written word has often been regarded as nearly sacred to Jews; the written name of God is seen as sacred, and so it was not to be disposed of in an ordinary manner (rather like the United States flag code's prescribing of how a worn-out flag is to be disposed of). Thus, unusable manuscripts (and later printed text) were stored in a geniza "until they could be given a proper burial." (Battles) The items found in a geniza might not just be religious writing, but just about any kind of written material which could possibly contain the name of God -- contracts, historical records, even children's copybooks.
The best known geniza is the one found in Cairo, Egypt, in 1890 when the Ben-Ezra synagogue there, built in 882, was being renovated. This particular geniza was relatively out-of-the-way, an attic room filled through a hole at top of a wall. One had to climb a ladder to reach the whole and deposit papers. Most genizas were ritually cleaned out at intervals, but this one was not emptied for a millennium.
The fragments found when the room was opened made their way into marketplaces and eventually came to the notice of Cambridge professor Solomon Schechter, who found that bits of Hebrew writing brought to him by Scottish women who had toured Egypt contained part of the book of Ben Sira (Ecclesiasticus) written about 200 B.C., a book of which early versions had only been seen in Greek before. Schecter went to Cairo and got permission to take as much as he wanted from the geniza's stores. He took only manuscripts, needing to put some limit on his acquisitions and figuring the last few hundred years' worth of printed books were not as interesting. He still took away more than 100,000 fragments of writing, and others later took more. These writings have provided incredible views of medieval Jewish (and Islamic) life in Cairo and the evolution of religious thought to researchers for more than a century. They even included previously unknown works by the poet Yehuda Halevi and the scholar/rabbi Moses Maimonides.
The physical fragments are located at various universities now, and several projects exist to put their transcriptions online. Still, perhaps as few as a third of the texts recovered from the Cairo geniza have been published.
Battles, Matthew. Library: An Unquiet History. New York: W.W. Norton & Company, 2003.