What is called the half-frame format is usually the 18x24mm format on regular 35mm film. It is the normal exposure format on 35mm movie cameras. For still cameras using the 35mm film, the usual format is 24x36mm, so still cameras taking 18x24mm exposures are called half-frame cameras.
There was a vogue of half-frame cameras in the 1960s, mainly from Japan, originating with the Olympus Pen models. It allowed to build a very compact camera, yet using the regular film that can be bought anywhere, unlike the other subminiatures that used exotic films (16mm, 9.5mm, etc.). This vogue ended when cameras like the Rollei 35 or the Olympus XA showed that it was possible to make cameras as small as the half-frame ones, but taking full 24x36mm exposures.
With a half-frame camera, you take twice the number of pictures that you would normally take with your roll of film. For example, you take 72 exposures on a 36-exposure roll, 48 on a 24-exposure one, and so on.
The most advanced half-frame camera that was designed as such from the start is the Olympus Pen F single lens reflex.
For some specific needs, there were cameras originally designed for full-frame pictures that were produced in very small series as half-frame models, for example some Leica, Nikon or Robot rangefinders, and some Alpa SLRs. They are mainly interesting as collectible.