A coherent state is a special quantum state that a coherent laser ideally emits [Wil07, p. 19].

That 👆 does not say much, but there is no straightforward way to define “coherent state”, a concept introduced by Schrödinger [SZ97, p. 46].

Mathematically and in short, a coherent state is the eigenstate of the positive frequency part of the electric field operator [SZ97, p. 46], but this requires definition of the electric field operator, which in turn requires discussion of the quantization of electromagnetic fields.

Classically, an electromagnetic field consists of waves with well-defined amplitude and phase, but in quantum mechanics, this is no longer true.

More precisely, there are fluctuations in both the amplitude and phase of the field [SZ97, Ch. 2].

A coherent state is a state that has the same fluctuations of quadrature amplitudes as the vacuum state but which possibly has nonzero average quadrature amplitudes [Van06, Sec. 4.6.1].