In mathematics and physics, continuous spectrum is, roughly speaking, a non-countable set of eigenvalues of an operator. An operator acting on a Hilbert space is said to have a continuous spectrum if its eigenvalues can be changed continuously. If the spectrum of an operator is not continuous, we say that it is has discrete spectrum. Some of the basic questions in spectral theory are to characterise the discrete spectrum and purely continuous spectrum, just as a measure, such as a probability measure, can typically be split into 'atomic' and 'continuous distribution' parts
The position operator usually has a continuous spectrum, much like the momentum operator in an infinite space. But the momentum in a compact space, the angular momentum, and the Hamiltonian of various physical systems, specially bound states, tend to have a discrete (quantized) spectrum. It is the reason why quantum mechanics was named in this way.
The quantum harmonic oscillator or the Hydrogen atom are examples of physical systems in which the Hamiltonian has a discrete spectrum. In the case of the Hydrogen atom, it has both continuous as well as discrete part of the spectrum; the continuous part represents the ionized atom.