What is FamilySearch? Where did it come from?

FamilySearch is a corporate entity entirely owned by The Church of Jesus Christ of Latter-day Saints. The name of the corporation is FamilySearch International. However the name "FamilySearch" is also the DBA (doing business as) of the Genealogical Society of Utah. I would guess that you could consider FamilySearch to be the successor in interest to the Genealogical Society of Utah which was established in 1894. The GSU is an incorporated, nonprofit educational institution entirely funded by the Church. Its headquarters are in Salt Lake City, Utah, USA with local representatives in all parts of the world.

Quoting from the FamilySearch.org Research Wiki:

In the 1990's, The Church of Jesus Christ of Latter-day Saints expanded the Genealogical Society of Utah and it eventually became known as FamilySearch. During the 1990's, Monte J. Brough, a General Authority and Executive Director of the Family History Department and president of the Genealogical Society of Utah, conceived of an Internet genealogy service which he proposed to church leaders. His ideas eventually developed into what today is known as FamilySearch.org. In 1999, the new website, FamilySearch.org, went live, providing an online medium for making genealogical records easily available to the public anywhere in the world at no cost.

Now quoting from FamilySearch.org Archives:

FamilySearch has partnered in this effort with more than 10,000 archives and more than 200,000 volunteers worldwide. Together, we create and preserve archival-quality images and indexes and make them freely available to millions of people looking for information about their ancestors. 

FamilySearch works together with records custodians to help them provide timely, economic, secure, and broad access to their records while maintaining access control and identity. The collections of FamilySearch include documents such as civil registration records; church records; and probate, census, land, tax, and military records. The collection also contains compiled sources such as family histories, clan and lineage genealogies, oral pedigrees, and local histories.

At the core of the activities of FamilySearch is their image acquisition and storage. Back in 1938 the GSU pioneered the use of microfilm and began filming records from archives and other repositories around the world. In 1998, FamilySearch transitioned to using digital imaging technology and began working on equipment that would digitized the 2.4 million rolls of microfilm stored in the Granite Vault in Little Cottonwood Canyon outside of Salt Lake City, Utah. In addition, the acquisition of records was transitioned to digital copies. Presently, hundreds of image acquisition missionaries are working around the world sending millions of digital images to FamilySearch for incorporation in the FamilySearch.org Historical Record Collections.

As records were uploaded online, there was a need to develop indexes of the information contained in the images. FamilySearch has sponsored a huge worldwide Indexing effort employing more than 100,000 volunteers. Over a billion records have been indexed so far and many millions more are in progress.

In October of 2013, FamilySearch announce strategic partnerships with three large online database companies, Ancestry.com, MyHeritage.com and findmypast.com. These partnerships will greatly expand the use of the FamilySearch Historical Record Collections. FamilySearch.org is a hugely popular and complex website with a tremendous variety of products supporting the genealogical community worldwide. One part of the website is dedicated to the Research Wiki, a free online genealogical resource with over 77,000 pages of helpful information about where and how to do research.

FamilySearch is also involved in the management and maintenance of the world famous Family History Library in Salt Lake City, Utah and over 4,600 FamilySearch Centers around the world.