Coherence is superficially rather simple, but it can be a difficult concept to understand. Coherence is the predictability of an oscillator—how long, in time and space, we can accurately calculate the state the oscillator will be in.
That's very simple, but oscillators turn up in the oddest of places. Let's start with something simple: a laser oscillator.
A laser can have three different coherences associated with it: time, length, and (confusingly) space. A laser emits light, which consists of an oscillating electric field. Ideally, once we have measured the field, it is always predictable but, in reality, this is not the case. Over time, the laser oscillator will drift, or even stop and restart. As a result, there is a limit to how far into the future the oscillator is predictable, called the coherence time. The coherence length is simply the distance that the laser light can travel in the coherence time. The spatial coherence of the laser beam is the predictability of the oscillator over the width of the laser beam at a frozen moment in time.
At heart, all coherence is this simple, but it turns up in unexpected places. A classic example is atomic physics. Imagine you have a cloud of atoms, all of which are sitting in the same ground state. We can shine a light that is resonant with an excited state of the atom on the cloud. As a result, some atoms absorb light and enter the excited state.
The combination of the light field and the atomic cloud form something very unexpected: an oscillator. The atomic cloud can store energy by absorbing photons and entering the excited state. The light field can also store energy in its electromagnetic field. So we have two energy storage systems that are coupled—energy can oscillate from one storage place to the other.
All the atoms start in the ground state. The illuminating light field drives them all into the excited state and then stimulates emission from the excited state, causing the atoms to decay to the ground state. If you measure the fraction of atoms in the excited state as a function of time, you get a nice oscillator. And, since it is an oscillator, you can define a coherence—this time the coherence is between the ground state and the excited state.
You might think that this system will remain coherent: in a fixed population of atoms, an atom is either in the ground state of the excited state. But an atom in the excited state might decay into some other, non-ground state, or interatomic collisions might cause a continuous rain of atoms out of the excited state. These sorts of processes cause the oscillations to decay over time, leaving an apparently constant fraction of atoms in the excited state and a constant fraction in the ground state—even though, individually, the atoms are still being excited into and decaying out of the excited state.
So, the coherence between the ground state and the excited state has a limited time, which we measure by looking at how fast these oscillations, called Rabi oscillations, decay away.
Coherence: a simple concept that, thanks to the madness that is quantum mechanics, turns up in some very unexpected places.