An edge witch is a Witch who deals with the edges- any edges, where boundary conditions apply. The edge between life and death, light and dark, good and evil, today and tomorrow, this world and the next... and many others. A witch who is described as an "edge witch" watches over and guards these edges. Edge witches learn to think fast, because edges can shift so quickly. It also hones senses that most people never know they have.
Both Granny Weatherwax and Nanny Ogg are skilled edge witches. In Thief of Time, Nanny Ogg is called by Wen to deliver Time's son (at different points in her life), and the senses she developed as a result of being an edge witch allow her to quickly understand the situation.
Granny Weatherwax continually demonstrates her skills as an edge witch. One example occurs in Carpe Jugulum, where she is able to successfully resist the bite of the vampire Count de Magpyr. It stuns the Count that anyone would have the willpower to resist, but Granny is able to do so, even though it costs her. She later teaches Tiffany Aching about being an edge witch.
Edge witches tend to look down on witches more concerned with the so-called High Magic (at least, as Wizards might define it), such as Letice Earwig. They instead praise witches who deal with helping people, such as Miss Level.
Some people perform similar roles to that of a witch without calling themselves a witch, and some of these can be edge witches too. An example is Sarah Aching.
Nac Mac Feegle Kelda perform similar roles to edge witches, as mentioned by a dying Kelda in The Wee Free Men. The Nac Mac Feegle, being less concerned with human right and wrong, have their own edges to take care of.