While Johnson-Nyquist noise is an equilibrium phenomenon, shot noise is a nonequilibrium effect, only present when there is a net current being driven through a system. Shot noise is a consequence of the fact that charge comes in discrete chunks. Remember, current noise is the mean-square fluctuations about the average current. If charge was a continuous quantity, then there wouldn't be any fluctuations - the average flow rate would completely describe the situation. However, since charge is quantized, a complete description of charge flow would instead be an itemized list of the arrival times of each electron. With such a list, a theorist could calculate not just the average current, but the fluctuations, and all of the higher statistical moments. This is called "full counting statistics", and is actually achievable under certain very special circumstances.
Schottky, about 90 years ago, worked out the expected current noise power spectral density, SI, for the case of independent electrons traversing a single region with no scattering (as in a vacuum tube diode, for example). If the electrons are truly independent (this electron doesn't know when the last electron came through, or when the next one is going through), and there is just some arrival rate for them, then the electron arrivals are described by Poisson statistics. In this case, Schottky showed that SI = <(I - < I >)2> = 2 e < I > Amps2/Hz. That is, the current noise is proportional to the average current, with a proportionality constant that is twice the electronic charge.
In the general case, when electrons are not necessarily independent of each other, it is more common to write the zero temperature shot noise as SI = F 2 e < I >, where F is called the Fano factor. One can think if F as a correction factor, but under sometimes it's better to think of F as describing the effective charge of the charge carriers. For example, suppose current was carried by pairs of electrons, but the pair arrivals are Poisson distributed. This situation can come up in some experiments involving superconductors. In that case, one would find that F = 2, or you can think of the effective charge carriers being the pairs, which have charge 2e. These deviations away from the classical Schottky result are where all the fun and interesting physics lives. For example, shot noise measurements have been used to show that the effective charge of the quasiparticles in the fractional quantum Hall regime is fractional. Shot noise can also be dramatically modified in highly quantum coherent systems. See here for a great review of all of this, and here for a more technical one.
Nanostructures are particularly relevant for shot noise measurements. It turns out that shot noise is generally suppressed (F approaches zero) in macroscopic conductors. (It's not easy to see this based on what I've said so far. Here's a handwave: the serious derivation of shot noise follows an electron at a particular energy and looks to see whether it's transmitted or reflected from some scattering region. If the electron is instead inelastically scattered with some probability into some other energy state, that's a bit like making the electrons continuous.) To see shot noise clearly, you either need a system where conduction is completely controlled by a single scattering-free region (e.g., a vacuum tube; a thin depletion region in a semiconductor structure; a tunnel barrier), or you need a system small enough and cold enough that inelastic scattering is rare.
The bottom line: shot noise is a result of current flow and the discrete nature of charge, and deviations from the classical Schottky result tell you about correlations between electrons and the quantum transmission properties of your system. Up next: 1/f noise.