Dust is a fictional form of dark matter (as we call it in our world), an elementary particle that is of fundamental importance to the novels. Dust is invisible to the human eye and can not be seen without the use of special instruments such as the The Amber Spyglass or a special film. However, while humans cannot see dust without the use of outside devices, creatures such as the mulefa are able to see dust with their own eyes.
Unlike ordinary particles, Dust is conscious. It falls from the sky and is attracted to people (especially adults) and objects made by people. This makes it of great interest to the Church, which believes that it may be the physical manifestation of Original Sin. We later learn that Dust actually confers consciousness, knowledge and wisdom. This allows creatures who have the ability to see dust to identify other sentient and intelligent creatures. An example of this is when the mulefa are able to distinguish Mary Malone as an intelligent being, because of the dust surrounding her, when compared to the other animals in the mulefa's world.
It is Dust that provides the answers given by the alethiometer, and by the I Ching system of divination.
Dust has various different names among the various different worlds within the trilogy. Dust was previously known (in Lyra Belacqua's universe) as Rusakov particles after their discoverer, Boris Mikhailovitch Rusakov. It is known also as Shadows in our world (Pullman relates Dust to Dark Matter), and the mulefa's word sraf accompanied by a leftward flick of the trunk (or arm for humans).
Angels, including The Authority, are formed when Dust condenses, but they are not in reality the human-like figures they appear to be.
Inspiration
There is a story in Genesis in which God creates Adam and Eve from dust, and in the first book of this series, Northern Lights, Lord Asriel shows the bible to Lyra: "In the sweat of thy face shalt thou eat bread, till thou return unto the ground; for out of it wast thou taken: for dust thou art, and unto dust shalt thou return". In chapter 12 verse 2, the letter to the Hebrews says,"let us throw off everything that hinders us, and the sin that clings so closely".
In Buddhism the term "dust" is almost universally used to refer to the sensation, knowledge and entanglement with the world that inhibits enlightenment.