In quantum mechanics, what exactly is a coherent state, and how does it differ from other states?
A coherent state is a state that is not an energy eigenstate, so the observables vary in time, but the uncertainty of the canonical conjugates (position and momentum) stays constant - unlike, for instance, in an ordinary wave packet moving in free space: this "floats away", the position uncertainty gets ever bigger and bigger.
A coherent state isn't the same thing as a squeezed state.
$$a|z\rangle = z|z\rangle$$
is a coherent state.
$$(\alpha x + i\beta p)|z\rangle = z| z \rangle$$
is a squeezed state.
For both, the product of uncertainties is minimized, but the former is more related to the Hamiltonian.