The original use of a cue card was to tell people on television what to say without it being completely obvious to the television audience that they were being prompted. (On radio, it made no difference if you had a script in front of you as long as you could consult it without being obvious; in theater, prompting was usually done vocally by a person in a prompt box in front of or at the side of the stage, though John Barrymore is said to have used visual cue cards at times.)

The person usually credited as inventor of cue cards was Barney McNulty, born 1923, who in 1949 was a page/usher at CBS on The Ed Wynn Show. At one point, Wynn was on medication which was making it difficult for him to memorize his lines for the live show, so Barney wrote them in marker on large sheets of cardboard from a hardware store near the studio, staying up most of the night to get all the star's lines down. Then he flipped the cards as needed during the show. Stan Freberg joked to him, "I can see it now...Barney McNulty, president of the Cue Card Corporation of America!" and McNulty ran with the idea, though he named his business "Ad-Libs" instead.

Flipping the cards at the right times, keeping them in order, and handwriting them legibly were important skills, especially when working with people who made last-minute script changes. When McNulty went on Bob Hope's tours to entertain U.S. military around the world, transport was required for 3,000 to 5,000 pounds of cue cards, and it was particularly important to put the right ones aboard the helicopters for each specific show. "Some of Barney's most vivid memories included trying to keep cue cards dry by placing them under airplanes in rainy tour sites, coping with frozen marker ink in Korea and having cards blow into the Pacific during a Hope performance on an aircraft carrier."

The teleprompter (originating in 1952, but taking a long time to spread) took the place of cue cards for many performers, but some people still preferred hand-written cards. Shortly before his death in 2000, McNulty worked on a television movie called "These Old Broads," featuring Elizabeth Taylor, Shirley MacLaine and Debbie Reynolds. And DejaMorgana pointed out that they are still widely used "in the rest of the world, especially on shows that don't have a lot of lead time (we used them a lot on Israeli talk shows). They are also used in interviews, when a producer wants the host to steer the interview towards something specific. And of course, there is the ubiquitous "Hurry It Up" cue used to rein in freestyling interviewers so that they can close a talk before the commercial break."

Over the years McNulty accumulated a collection of more than 100,000 of his cue cards, noting in 1983, "I want to keep them because it's a continuing history of show business."

The description "cue card" has spread to many other situations to describe a portable written reminder of information. A lot of cue-cards these days are tear-out pages printed on thick card stock and found in reference books, or pre-laminated cards sold to give you the highlights of some procedure.

Sources:
Hale, Lee, with Richard D. Neely. Backstage at the Dean Martin Show. Dallas, Texas: The Taylor Publishing Company, 2000.
http://www.letsmakeadeal.com/barney.htm
http://www.tvacres.com/commun_signals.htm