In short, a veil is a 'subsystem mask' i.e. smoke and mirrors or a construct/fragment so that the whole subsystem can "pass" as one system member.
One of the best anecdotal descriptions of veils must come from When Rabbit Howls by The Troops for Truddi Chase. "The Woman" is the main front for their body, and she is distressed over the incoherence and fragmentation of her life. She feels hollow, vapid, but often feels "filled" suddenly with presence and direction, and doesn't fully feel like she owns the chaos of her life. Her memory is nearly non-existent. She lives from to do list and carefully guarded routine to list or routine. In fact, she has a purse full of lists of routines to follow.
She can't explain any of it, and she decided she wants help.
The Woman is a veil. As the story goes on, you find out that she is still sentient, but essentially without direction, motivation or affect (emotions) unless another Troop memeber is "filling her". She's essentially a hand-puppet, an automaton, but with the curse of sentience -- a sense of "I" in and of herself, even though she is essentially hollow.
Veils are mental constructs infused with a sense of sentience, and worn by internal system members as a spacer from the external world. No one needs to directly front if they use a veil. And the good part about the vapid hollowness of a veil is that they forget easily, they don't notice being used, they are ephemeral in that their existence is ghostly, so they don't remember why they were just angry, or sad, or what they just ate.
Veils are not only for fronting to the outside world. Sometimes they are like gatekeepers -- instead of a solid presence between subsystems, there is a puppet so that the subsystem has a steady face to the other system(s) they interact with. Thinking of the boy in the bubble movie from the 70s -- it's like a film between the subsystems, to keep what's in one in one, and what's in the other in the other. It also means that people in one subsystem might not suspect that there is another subsystem behind that veil. The veil appears to be a person within their subsystem. But instead it's an airlock or an interface that really honestly thinks it's a member of the subsystem.
A veil is similar to a mask, but with sentience. Of itself it is able to say "I" and how much more it can say beyond that is probably dependent on the particular veil (as these things are probably spectrums or continuums rather than discrete and specific points).
A veil may be much more than just a "mental construct" or a "mask with sentience". That approach can belittle the importance of veils and the importance and reality of the people who are those veils. (How would you like to be called a mask with sentience?") A system's veil may even be it's OriginalPersonality.
Multiples from the older generation refer to veils as shells.
Shells added by Bluejay Young