What is a contact sheet in photography and what's its purpose?

Until recent times, most all photo images were made using a film camera. Photo film naturally produces a negative image. Some exceptions noted however, the vast majority of photo images were made on negative film. The resulting negative displays all tonal relationships as reversed. What is dark and what is light in reality is reversed as to tone on the film. Negatives are a means to an end, they themselves are unusable as an image however they are subsequently converted into positive images for our enjoyment.

It is common practice to convert negatives into positive prints by re-exposing them onto photo paper. This re-exposure can be made on an enlarger by projection or; by placing the negative in direct contact with the photo paper in a darkened room. The negative is held flat to the photo paper by sandwiching it atop the photo paper under a cover of plate glass. Once the negatives are positioned on the photo paper, the room lights are momentarily turned on. This action exposes the negative images onto the photo paper. The now exposed paper is developed by bathing the paper in a series of chemical baths. The result is a positive image called a “contact” print.

In the early days of film photography, the cameras as well as their film were quite large. In many cases, a contact print was sufficient in size thus they were hung on the wall or placed in an album. As technology marched on, films improved. Modern cameras and their film became miniaturized. These modern film camera’s accepted roll film. The developed roll of film is strip of film that contains many images.

Photographers cut these long rolls of developed film into manageable strips. Often a strip contained 4 or 5 miniature negative images. Because the images are negatives, they are hard to judge as to acceptability. It is common practice to arranged several strips atop photo paper held in place by a sheet glass cover. The room lights were turned momentarily. This method exposes numerous miniature negatives on a single sheet of photo paper.

These contact sheets are inexpensive to make and they allow the photographer to edit out unwanted images and thus choose those images to enlarge and present to the client.

A contact sheet is the conversion from negative to positive without enlarging. This way you can print a complete film on sheet of photo paper quickly.
Then, with the help of a loupe, especially with 35mm film, you chose the ones of which you're gonna enlarge and do a final print of.
With medium format it was also a quick way to get useable prints very quickly, especially snap shots, as the negatives used to be large enough. Quite popular when photography was VERY expensive, I have quite a number of contact prints from when I was a boy in the 1960ies.

So the purpose was to get cheap and easy (B&W) prints either for editing or for final use.

Here is an example already marked for enlarging:
http://2.bp.blogspot.com/-v5fMqtqJ6zg/UZEMWlAPgzI/AAAAAAAAe7E/Pk955llJ2x4/s1600/290039_409269655783015_388561536_o.jpg

Nowadays the term is also used to print a number of chosen digital pics on one sheet of paper.

That applies with film negatives. Instead of enlarging the picture, the paper is exposed in close contact to the (cut) film strip so that all frames in the roll get printed in a (usually) 8x10" (8R) sheet. Photographers, with the use of a loupe (magnifying glass), can then assess if each frame is worth printing or not.