The pinhole in a pinhole camera acts as the lens. The pinhole forces every point emitting light in the scene to form a small point on the film, so the image is crisp. The reason a normal camera uses a lens rather than a pinhole is because the lens creates a much larger hole through which light can make it onto the film, meaning the film can be exposed faster.

The smaller the hole is the sharper the image will be but the dimmer the projected image will be. Ideally, the size of the aperture should be 1/100 or less of the distance between it and the screen.

Usually, you have to expose the film for a long time because the pinhole lets so little light through. Typical exposures range from 5 seconds to hours or sometimes even days. A manual shutter is usually used.

One common use of a pinhole camera is to capture the movement of the sun over a long period of time. This is referred to as Solargraphy.

Images can be projected onto film or a charge coupled device, or they can be projected onto a translucent screen for real-time viewing (also known as a camera obscura). This is one popular way for viewing solar eclipses.

The 10th century Arab physicist, astronomer and mathematician, Ibn al-Haytham (Alhazen) invented the first pinhole camera after noticing the way light was streaming through a hole in a window shutter. He improved on the camera after realizing that the smaller the pinhole, the sharper the image (though the less light). He designed the first camera obscura. But it was not until 1850 that a Scottish scientist by the name of Sir David Brewster actually took the first actual photograph with a pinhole camera.

Kodak provides easy to follow directions how to make your own pinhole camera.

Sources: wikipedia, How Stuff Works, Kodak