These Bibles are placed by a society called "The Gideons International." They take their name from the Biblical figure Gideon, in the book of Judges chapters 6-8; broadly speaking, the account speaks of Gideon as an Israelite leader who helped his people throw out their Midianite oppressors and eliminate worship of idols. A particular element of the story is Gideon's heavily outnumbered force of 300 soldiers defeating a larger enemy, through craftiness and divine guidance and favor. He is also mentioned in the New Testament, Hebrew 11:32-34, as one of the great Old Testament heroes of faith.

Why this society chose him for their namesake is, alas, not totally clear to me; their official story is that it came to them in prayer. The society was founded in 1899 as an association of (Protestant) Christian businessmen, and that is what it still is today. They're denominationally unaffiliated but their statement of faith is broadly evangelical Protestant; clergy are excluded from membership, but members must be recommended by their minister. Their primary mission quickly became the distribution of Bibles - your question suggests that hotels use their Bibles, but the drive to place them comes from the Gideon society rather than from the hotels. They also hand out Bibles, if the local law and policy allow, at or near schools, distribute to hospitals, prisons, and military personnel, and help distribute them to churches overseas. One pop culture reference is from the 1950 Broadway musical Guys and Dolls, where the main character, a gambler, suprises the Salvation Army missionary by accurately quoting the Bible - which he credits to reading Gideon Bibles in hotel rooms. When we say "Gideon Bible," it's not a special Gideon translation; traditionally they used a King James translation, but now sometimes offer other versions.

Theologically, this effort is linked to the Christian belief that the Bible is the word of God, and the Protestant conviction that the Bible is the only infallible and only necessary source of information about spiritual matters. While other Christian traditions put higher emphasis on tradition, church hierarchy, sacraments, or other devotional practices, American evangelicalism has a stronger individualist emphasis ("my religion is between me and God, and I don't need churches or pastors to tell me what to believe"), the need to make a decision to accept Jesus as savior, and have an experience of conversion (as opposed to a merely intellectual assent to doctrines). Many Gideon-provided Bibles will include some material about salvation according to this evangelical scheme, even guiding one through how to say a "sinner's prayer," make a decision for Christ, and be saved.