Why do we use the electron volt?
Why did it come to be the electron volt and not, say, just a prefix of the joule, like the nanojoule?
Does the electron volt represent anything particular as far as the mathematics goes? I am guessing that it does, and if so, what is it that the electron volt exactly represents in terms of the mass of a particle, as I have seen it used for both the energy of a photon and the mass of subatomic particles?
The electron-volt is a convenient unit of energy when considering electrons moving between points at different potentials. The convenience came from having numerical values which are around or greater than one, $1 \rm eV = 1.6 \times 10^{-19} \rm J$. It was first used in the 1930s.
So one perhaps has a better "feel" for the difference between 1 and 100 eV than $1.6 \times 10^{-19} \rm J$ and $1.6 \times 10^{-17} \rm J$ and the value in electron volts is easier to write.
Electron energy levels are conveniently quoted in electron-volts and then nuclear energy levels in MeV show a clear difference in terms of scale.
Then using eV/c² with the appropriate prefix as a unit of mass also becomes convenient; e.g. the mass of the electron as 500 keV/c² and that of the proton as 1 GeV/c².
It is not an SI unit but is retained because as well as being convenient it was and still is in widespread use in the scientific community.
Addressing only why it is used/useful in science today, not why or how it came about
The other answers seem to come from a particle physicist's point of view; for a chemist the electronvolt is convenient as well:
Please note these are "order of magnitude" ranges.
"Historically, the electronvolt was devised as a standard unit of measure through its usefulness in electrostatic particle accelerator sciences because a particle with charge $q$ has an energy $E = qV$ after passing through the potential $V$; if $q$ is quoted in integer units of the elementary charge and the terminal bias in volts, one gets an energy in eV."
source
Further, you will have to admit that energies written as $x \cdot 10^{-19} \textrm{J}$ are not the most useful numbers to work with. Using an arbitrary standard quantity (e.g. Angstrom) as comparison to obtain numbers that do actually mean something to us and that make it easier for us to talk about them is quite a widespread custom in physics.
It is just a convention, and not a particularly convenient one. In particle physics we hardly ever use $\mathrm{eV}$; it is much more common to use $\mathrm{MeV}$, $\mathrm {GeV}$ or even $\mathrm{TeV}$. The electronvolt scale is not a natural scale for particle physics: typical energies are, at least, a million times higher than that (except for neutrinos). We use it for historical reasons, but I guess you'll agree that it is more convenient that Joules: an electronvolt is somewhat closer to the mass of a proton than a Joule ($m_p\sim 10^{-10}\ \mathrm{J}\sim10^9\ \mathrm{eV}$).
Perhaps I should add that in solid state physics electronvolts are sometimes a natural scale.
Originally, eV might have been the right unit for electron energy used by people who were doing experiments with cathodic tubes. In those experiments, a cathode was emitting electrons if there was a cathode-anode bias. The multiples of eV are the right unit if you do accelerator physics. MeV, GeV, TeV are chosen because they are also closest to the order of magnitude of the electron energies in those accelerators. So, as a rule of thumb, one chooses the unit closest to the order of magnitude of the energy in the type of physical phenomenon of interest.
For example, if you do electron transport in nanostructures (like carbon nanotubes) you might want to use meV for energies, nm for distances, and fs for time. If you are interested in the band structure of solids, the best unit is eV, as the band gaps for insulators are usually a few eV's.
On the other hand, if you do engineering and you work mostly with macroscopic objects, you will work with SI units.
