What is usually called a discrete simplicial set?
I know, though, what is a discrete topological space.
It is used here, on page 15, but never defined. They say on page xiv, that "the term space will be often used interchangeably for topological spaces or simplicial sets."
A discrete simplicial set is a simplicial set with no nondegenerate simplices except in degree $0$. Equivalently, it is a simplicial set in which all the face and degeneracy maps are isomorphisms, or a simplicial set which is isomorphic to the singular simplicial set of a discrete topological space, or a simplicial set which is isomorphic to a constant functor $\Delta^{op}\to \mathtt{Set}$.