My understanding is that an electron volt is the energy an electron needs to move against an electrical gradient of 1V (imagine like rowing against the current). It is a unit of energy, so at first glance it doesn't really make much sense to use it a a mass unit. However, since e = mc2, you can calculate the mass that is equivalent to 1 eV (which, in kg, turns out to be about 10 to the power of -36).
The electron volt is related to the electric charge of the electron, not to its mass. Similarly, a magnet weights more than its strength turned into mass.
The sibling comments are good, but let me add my own version.
eV is a unit of energy, and eV/c^2 is a unit of mass. You can drop the c^2 if you are lazy, and if someone ask you can say that you using a system of units where c=1 and ħ=1. This is fine in the second half of a Physic degree, but you can get in trouble for forgetting the c^2 in the first half or during secondary school.
The technical definition is in a sibling comment, but an eV is similar to the energy that an electron gets or loose when it makes a jump inside a molecule or between two molecules. The exact number depends on the molecule and the jump, it may be x10 bigger or smaller.
For example if you connect a led to a battery, each electron gets like 1.5eV when it pass through the battery, and release that energy as a photon of light in the led. If you ignore a lot of technical details, the electron makes one or two jumps in the battery and a jump in the led, but this is a huge oversimplification.
Note that a led releases like 10^18 photons per second, so each photon with approximately 1eV has a tiny amount of energy.
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On the other hand, if you destroy an electron you get 511000eV of energy. You can't destroy a single electron, but you can make an electron colide with a positron that has the same mass and in total you gen 511000eV+511000eV of energy. So destroying an electron release much more energy than an electron that jumps inside a molecule.
In special relativity there is an equivalence between energy and mass, so if the energy you get while destroying an electron is 511000eV then the mass is 511000eV/c^2. You can also measure the mass directly without destroying the electron, and you get the same number (probably with a bigger uncertainly, I'm not sure how they measured so many digits).
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As an analogy, if you move around and climb some steps and make some jumps, each big movement absorbs or release approximately 1000J.
If someone makes a antimatter clone of you, and you both colide, your mass will be released as energy and they will get like 10^18J+10^18J = 1000000000000000000J+1000000000000000000J.
Changing the units, your mass is like 510^37eV/c^2 and if you colide with your antimatter evil clone, you both will release 510^37eV+5*10^37eV of energy.
Because eV is not (directly) a mass measure. Though of course it can translate into a mass. To C&P Wikipedia it's "the measure of an amount of kinetic energy gained by a single electron accelerating from rest through an electric potential difference of one volt in vacuum."
electron volts is a unit of energy. Using E = mc^2, one expresses masses in the equivalent energy quantity. It turns out that electron-volts is a convenient scale for the (energy equivalent) mass for elementary particles -- much better than trying to express the mass in kilograms.
Or energy in Joules, which are around 10^19 bigger. So you'd have the inverse of that everywhere as an inconvenient factor.
As it happens eV works nicely from meV up to TeV for particles in all the most useful contexts, from chemical bonds to particle experiments. You only start getting insanely huge factors when you stop being subatomic.
It's a convenient size unit for particles! Also energy is natural when the energy of the particle can vary.
Also, the unit used for mass of particle is usually eV/c² or MeV/c² which is more correct (unless in c=1 units) but the c part is sometimes dropped by convention.
(also i would have expected an electron to be... 1 electron volt?)