A solarigraph, sometimes called a solargraph, is a very long exposure which shows the path of the Sun. Often over the course of a single day, a few days, or even six months from one solstice to the other. To make one, a piece of photographic paper (the darkroom kind) is loaded into a camera and left alone. Often the camera is a cheap, home made pinhole style camera, so not much is lost if the camera is damaged, lost, or stolen.
When left in place for many hours, an image will form on the paper (without being developed). The Sun's path will be traced out, and a faint image of the foreground will also be visible. People often make exposures for one day, or six months (from one solstice to the other), though some have been left for a year or more. How long to leave it is a choice up to the phogorapher as a creative choice, the more days left out, the more tracks of the Sun that will be captured.
Once the desired exposure length has been reached, the paper is removed from the camera. This should be done out of direct sunlight, but total darkness is not necessary. The paper is not developed, no chemicals are used. Instead the paper is either scanned or photographed. The paper will continue to fade over time since it was never "fixed" (and cannot be), so the digital photo is the end result, and may be processed to bring out faint details, or invert the image.
The Wikipedia entry for solarigraphy is a good start.
Reddit has two subredits one called solargraphy the other solarigraphy for discussion, questions, and sharing photos. You may find solarigraphs also common in many other solar imaging forums. There is also a Flickr group called solargraphy.
3D printable solarigraph cameras designed to hold either 5x7 or 8x10 paper: Camera
3D printable can holders, for holding the "can" style solarigraphs securely for long periods: Holder
Some other camera boxes that accept full sheets of 5x7 or 8x10 paper.
One of many videos showing how to make a solarigraph camera out of a soda can.
The type of paper used for a solarigraph is the type of photographic paper used in a darkroom. Unfortunately search engines will tend to favor the more common "photo" paper used in inkjet printers. Below are some photos of the boxes and trade names which may help searching. Any darkroom paper should work fine, but these are the ones I have tested. Each produces different colors, which I plan to document here eventually.
I avoid glossy papers, and recommend you use matte if you can. The glossy papers can cause reflections that produce multiple images of the Sun.