In my opinion, the most intuitive way of explaining the meaning of harmonic oscillator coherent state is the following:
An Harmonic Oscillator Coherent State (AOCS) is a solution of the time-dependent Schodinger equation for the quantum harmonic oscillator. This means that the probability density distribution, $|\psi|^2(x,t)$ evolves in time, i.e. it is not a stationary state. Do you remember the ground state of an harmonic oscillator?! The famous $|0\rangle$ state corresponds to a Gaussian probability density, centered in the origin. The particular feature of this state is that the uncertainity on the position, $\Delta x$ and the uncertainity on the momentum $\Delta p$ minimize Heisemberg's uncertainity principle. Eigenstate $|0\rangle$ is a stationary state and the associated probability density $|\psi|^2(x)$ doesn't depend on time.
Well, an Harmonic Oscillator Coherent state can be visualized as an "oscillating" $|0\rangle$ state. In other words, an AOCS is a gaussian wavefunction which oscillates back and forth, around position $x=0$. But at every instant the shape of the gaussian is always the same (and minimizes the uncertainity principle). So, some quantities like $\langle x \rangle $ (which corresponds to the peak of the Gaussian wavefunction), $\langle p \rangle $, (which can be associated to the velocity of the gaussian's peak) have a sinusoidal time evolution. That's why they often go under the name of semi-classical quantum states! In other words: if the amplitude of oscillations is $\gg$ $\Delta x$, you re-obtain the classical equations of motion for a point-like massive particle subjected to an elastic force.
Of course you have many possible coherent states. To be more precise, coherent states form a 2D planar manifold. This means that, in order to build an AOCS you have 2 degree of freedom: amplitude and phase of the oscillation of the gaussian wave packet.
They are coherent in the sense that the phase of the various chromatic components is arranged in such a way to reproduce this oscillating behaviour.