The total energy of a mode in a quantum mechanical resonator is given by $E_n ~=~ (n+ 1/2)hf$ where $n$ is the number of modes. So when there are no modes or vibrations, i.e. $n=0$, the energy is called the zero-point energy.
What I don't understand is, if there are no modes, then what is this energy associated with? So what exactly is a mode? Also, is there a way of measuring this zero-point fluctuations? Why is the 1/2 photon introduced?
The zero-point energy for the quantum mechanical harmonic oscillator can be related to the Heisenberg uncertainty principle (HUP). A bit oversimplified, the point is intuitively that if the mechanical energy
$$H~=~\frac{p^2}{2m}+\frac{1}{2}m\omega^2 q^2, \qquad \omega ~:=~ 2\pi f,$$
is zero, then the position $q$ and the momentum $p$ must also both be zero, in contradiction with the HUP. Instead both $q$ and $p$ "fluctuate/are fuzzy" in the ground state, thereby producing the zero-point energy.
The zero-point energy is e.g. important to theoretically explain the Casimir effect, which in turn has been experimentally observed.
In cosmology, zero-point energy is related to the cosmological constant.
Not really as a direct consequence of the HUP. Consider: From Q.F.T we know there are fields which exist everywhere in spacetime, and all particles are higher-energy states of these fields. E.g., there's only one electron field which exists everywhere, and every electron is actually an excitation of this field. Now, when you think of a vacuum you usually think of "empty space". I.e., no particles. But the fields are always there even when there are no excitations to higher-energy states, creating particles. So a vacuum without any particles still has fields. When fields are not in an excited energy state, they are in a state of lowest energy. This state is the only state in which there are no particles, and is also known as the ground state. However, this is a state of lowest energy, it is not a state of zero energy. There is energy that the field always has, even when it's in its' ground state. This energy is called the vacuum energy or zero-point energy. I.e, A vacuum is not really empty, even if there are no particles. There are still fields. The vacuum energy is simply the energy that fields have when they are in the vacuum state.
It's associated with the confinement of the particle. As the coefficient of the quadratic term in the harmonic potential goes to zero, the particle becomes less & less localized and the the zero point energy drops accordingly. It can be as small as you want.
When you speak of photons, now you're referring to QFT and the energy in a mode of the EM field. There's no 1/2 a photon. There's an arbitrary normalization of the zero point of the vacuum field energy but no one knows how this relates to the cosmological constant -- in QFT it's arbitrary, and there are only semiclassical arguments of questionable rigor for how it should couple to the curvature tensor in GM.
Incidentally, you can also derive the Casimir effect without any reference to zero modes of the field, as the analog of the van der Waals force between polarizable atoms/molecules. The wikipedia page has the details.
Choice of the zero-point energy is similar to choice of zero of the potential - it is rather arbitrary. One can safely put it equal to zero. However one should not understand it as a "zero-motion". The motion in the ground state exists (HUP) and it determines the size of systems created by bound particles (like atomic size, etc).
