A point s belongs to the discrete spectrum of an operator T, if you can find a neighbourhood of the point s that contains no other points of the spectrum of T. For example the spectrum of the negative Laplacian is the set [0,\infty) which has no discrete part. But there are examples of operators T whose spectrum contains also discrete points. For example, if T is the Hamiltonian of the hydrogen atom i.e. -Δ-1/|x| then its spectrum σ(T) has a discrete set of eigenvalues of the operator T.
