In classical optics light is thought of as waves radiating from a source, and coherent light is thought of as the light from many such sources that are in phase. For instance a light bulb gives off light that is actually the result of the light being emitted at all the points along the filament, and such light is not coherent because the process is highly random. In the laser however, the light is emitted "all at once" from a carefully controlled mixture of atoms, the process is not random and the resulting light is highly ordered, or coherent.
In the quantum model we understand light to be somewhat more complex than a simple wave, including, among others, the properties traditionally measured in classical optics. However the quantum model describes all of nature using this description, with everything from light to solid objects sharing similar behaviours. Under the quantum model then, the concept of coherence takes on a broader and more general meaning that can be applied to any set of particles in the proper state.
In the wave-particle duality reminiscent in all quantum states, the coherent state stands on one extreme, namely, it behaves as much as possible as a wave, whereas the Fock state (like a photon) stands on the other end, namely the particle behaviour. In fact neither the particle or wave are "real" in this model, but they are useful fictions for describing the overall behaviour of systems, and in this case any system can display properties that make it easier to consider as wave-like or particle-like.
In quantum optics the coherent state is the quantum state of light emitted by an ideal laser. In condensed matter physics it describes fields with coherence such as Bose-Einstein condensates.
In a Fock space, the coherent state is the eigenstate of the annihilation operators:
In the occupation number formalism (see Fock space and quantum harmonic oscillator), the coherent state becomes the eigenstate of the annihilation operator of a particle in its ground state, written just a. Let us call |α> this coherent state, so that by definition it formally reads:
The coherent state does not display all the nice mathematical features of a Fock state, for instance two different coherent states are not orthogonal:
Another difficulty is that a† has no eigenket (and a has no eigenbra). The formal following equality is the closest substitute and turns out to be very useful for technical computations: